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FACTORING 1 College Algebra with Trig Name:__________________________________ Lesson/HW- Factoring Polynomials Date:___________________________________ Objective: factor polynomials Factor completely. If the polynomial cannot be factored, write simplified. (1) 8x – 24 (2) xy – 17y (3) x2 – 169 (4) x2 – y2 (5) x2 + y2 (6) 3x3 – 3x (7) 9x2 – 36y2 (8) 2x3 – 4x2 – 6x (9) 5x2 – 13x + 6 2 (10) x2 – 6x + 2 (11) 4a2 + 12ab + 9b2 (12) 36w2 – 16 Factor completely. If the polynomial cannot be factored, write simplified. (13) 6 – 5x + x2 (14) 40 – 76x + 24x2 (15) 2x2 + 4x – 1 (16) 2x2 + 28x – 30 (17) 6x2 + 7x – 3 (18) 18x2 – 31xy + 6y2 3 College Algebra with Trig Name:__________________________________ Lesson/HW- Factoring Special Polynomials Date:___________________________________ Objective: factor special polynomials Guidelines for Factoring: (1) Factor out the Greatest Common Factor (GCF) – “Gotta Come First” Monomials (2) Factor Binomials – check for special products, for any numbers a and b: (a) Difference of Two Perfect Squares: a2 – b2 = (a + b)(a – b) (b) Sum of Two Perfect Cubes: a3 + b3 = (a + b)(a2 – ab + b2) (c) Difference of Two Perfect Cubes: a3 – b3 = (a – b)(a2 + ab + b2) (3) Factor Trinomials – check for special products, for any numbers a and b: (a) Perfect Square Trinomials: a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 (b) General Trinomials: acx2 + (ad + bc)x + bd = (ax + b)(cx + d) (4) Factor Polynomials – if there are four or more terms, try factoring by grouping. Factor completely: (1) x3 + 27 (2) x3 – 64 (3) 27x3 – 8 (4) 2x3 + 16 (5) x3 – 4x2 + 3x – 12 (6) x3 + 5x2 – 2x – 10 (7) 5a2x + 4aby + 3acz – 5abx – 4b2y – 3bcz 4 Factor completely: (8) 35x3y4 – 60x4y (9) 2r3 + 250 (10) 100m8 – 9 (11) 3z2 + 16z – 35 (12) 162x6 – 98 (13) 4m6 – 12m3 + 9 (14) x3 – 343 (15) ac2 – a5c (16) c4 + c3 – c2 – c (17) ax – ay – bx + by (18) 64x3 + 1 (19) 3ax – 15a + x – 5 5 College Algebra with Trig Name:__________________________________ Lesson/HW- Operations with Rational Expressions I Date:___________________________________ Objective: perform operations with algebraic fractions and simplify mixed expressions Do Now: Factor completely. If the polynomial cannot be factored, write simplified. (1) 3x2 + 10x + 8 (2) 8x3y6 + 27 (3) 4x2 – 12x + 5 (4) 4x2 – 4x – 48 Showing all work, perform the indicated operation and simplify your answer. x2 9 4 x 2 20 x x 2 3x x 2 5x 6 (5) (6) x 2 x 20 4 x 2 12 x 2x 2 x 6 x2 4 6 Showing all work, perform the indicated operation and simplify your answer. 12a 4 b3 8x 8x 16 (7) (8) b 12 2x 8 2 32x 2 3 x 2 12 3 x 2 15 x 18 9x 45 3x 3 (9) (10) 2 2x 2 x 6 x 2 3x x 4x 5 2 x 1 x 3 x 2 x 2 2x 1 2x 2 x 6 4 x 8 (11) (12) 3x x2 1 4x 2 9 6x 9 7 Showing all work, perform the indicated operation and simplify your answer. a2 64 8a x2 1 3x 3 (13) (14) 2 8a 64a 9a 72 2 3x 2 x 2x 1 2c 2 c 6 12 x 2 10 x 25 x 2 25 (15) (16) 4c 8 6c 9 x5 5 x 25 (17) 2x 3 10 x 2 8 x x 2x 8 2 2x 2 x 1 (18) 2x 2 x 6 2x 2 4x 2 9 8 College Algebra with Trig Name:__________________________________ Lesson/HW- Operations with Rational Expressions II Date:___________________________________ Objectives: perform operations with algebraic fractions and simplify mixed expressions simplify mixed expressions and complex fractions Do Now: Factor completely. If the polynomial cannot be factored, write simplified. (1) x2 + 2x + xy + 2y (2) 64x2 – 676 (3) 1 – 125y3 (4) 3a2 – 2b – 6a + ab Showing all work, perform the indicated operation and simplify your answer. 2y 8 7k 2 8k (5) (6) y 16 2 16 y 2 4k 3 3 4k 9 Showing all work, perform the indicated operation and simplify your answer. 1 5 x 2 (7) x (8) x 1 x 5 1 x 3 6 2x 1 1 4n 2 9 (9) (10) x x 2 x 2n 3 3 2n 10 Showing all work, perform the indicated operation and simplify your answer. 3a 1 1 3y 4 y2 (11) (12) a 1 2 a 1 5 4 x4 x4 x2 2 (13) (14) x 4 x 16 2 x4 11 Showing all work, perform the indicated operation and simplify your answer. 2a 5 1 2b 1 1 (15) (16) a 5a 6 2 a3 b b 12 2 b4 1 x z x (17) 3 (18) z x 1 1 1 3 x z x 12 College Algebra with Trig Name:__________________________________ Lesson/HW- Complex Fractions and Equations Date:___________________________________ Objectives: simplify mixed expressions and complex fractions solve equations with algebraic expressions solve real-world applications with algebraic expressions ON A SEPARATE SHEET OF PAPER, ANSWER EACH OF THE FOLLOWING QUESTIONS SHOWING ALL WORK! Perform the indicated operation and simplify your answer: 5 6 m5 1 a 2 b 2 y 1 m y y2 (1) (2) y 1 m3 (4) ab ba y (3) 1 3 m 1 y Solve each of the following equations and check: x 16 1 1 1 4 (5) 2 (7) 2 x 8 x 64 x 8 2b 6 2b 6 b 9 1 1 6 x 1 16 (6) 2 (8) 2 h 1 h 1 h 1 2x 8 x 4 x 16 Show All Work: (9) The area of a rectangular patio is represented by the expression (6x2 + 13x – 5). The width of the patio is (3x – 1). Write a simplified expression to represent the length of the patio in terms of x. 3a a 2 (10) If the length of a rectangular field is represented by the expression 2 , and the width is represented a 9 a 2 a 12 by , what simplified expression represents the area of the field? a4 13 Unit 1: Algebraic Fractions, Equations & Factoring Definitions, Properties & Procedures Factoring the process of writing a number or algebraic expression as a product Least Common the least common multiple of two or more given denominators Denominator (LCD) has the same properties as a numerical fraction, only the numerator and Algebraic Fraction denominator are both algebraic expressions Rational an algebraic expression whose numerator and denominator are polynomials and Expression whose denominator has a degree of one or greater reducing or simplifying a rational expression means to write the expression in lowest terms, which can only be done with a single fraction, a product of fractions Simplifying or a quotient of fractions. If there is an addition or subtraction sign in the Rational numerator (or denominator), it must be factored first and then like factors with the Expressions denominator (or numerator) can be canceled. Note: you cannot reduce across a sum or difference of two or more fractions! To multiply rational expressions: (1) Factor each numerator and denominator completely (2) Cancel any like factors in any numerator with any like factors in any denominator Multiplying & (3) Multiply the remaining expressions in each numerator Dividing Rational (4) Multiply the remaining expressions in each denominator Expressions (5) Reduce if possible To divide rational expressions: (1) Multiply the first fraction by the reciprocal of the second fraction (KCF) (2) Follow the steps above to multiply rational expressions (1) Find the least common denominator among all fractions (if necessary) (2) Multiply each denominator by an appropriate factor to make it equivalent to the Adding & LCD; and multiply each numerator by the same factor that you multiplied its Subtracting denominator by (multiply by a “fraction of one”) Rational (3) Combine all numerators (make sure the signs are placed appropriately) and Expressions simplify; and put over LCD (4) Reduce if possible a fraction that contains one or more fractions in the numerator, the denominator, or both Complex Fraction To simplify complex fractions: Combine fractions in the numerator and denominator separately by adding or subtracting. Once there is a simplified fraction above a fraction, use the steps for dividing fractions to further simplify the expression. an equation that contains one or more rational expressions To solve rational equations: Rational Equation (1) Find the LCD (2) Multiply each fraction by this LCD (3) Cancel all denominators (4) Solve the remaining equation for the given variable Greatest Common the product of the greatest integer and the greatest power of each variable that Factor (GCF) divides evenly into each term 14 a polynomial of the form a2 – b2, which may be written as the product (a + b)(a – b) To factor a difference of two perfect squares: Difference of Two (1) Create two empty binomials ( )( ) Perfect Squares (2) Take the square root of the first term of the given binomial and put it in the 1st position in each binomial (3) Take the square root of the last term of the given binomial and put it in the 2nd position in each binomial (4) Make one binomial a sum and the other binomial a difference Sum of Two Perfect Cubes: a polynomial of the form a3 + b3, which may be written as the product (a + b)(a2 – ab + b2) To factor a sum of two perfect cubes: (1) Create an empty binomial and an empty trinomial ( )( ) (2) Take the cube root of the first term of the given expression (a) put it in the 1st position in the binomial Sum of (b) square it and put it in the 1st position of the trinomial Two Perfect Cubes (3) Take the cube root of the last term of the given expression & (a) put it in the 2nd position in the binomial (b) square it and put it in the last position of the trinomial Difference of Two (4) Find the product of the terms in the binomial and put it in the middle position Perfect Cubes of the trinomial (5) Arrange the signs as follows: ( + )( − + ) Difference of Two Perfect Cubes: a polynomial of the form a3 – b3, which may be written as the product (a – b)(a2 + ab + b2) To factor a difference of two perfect cubes: Follow above steps and arrange the signs as follows: ( − )( + + ) a trinomial whose factored form is the square of a binomial; has the form a2 – 2ab + b2 = (a – b)2 or a2 + 2ab + b2 = (a + b)2 To factor a perfect square trinomial: (1) Create two empty binomials ( )( ) Perfect Square (2) Take the square root of the first term of the given trinomial and put it in the 1st Trinomial position in each binomial (3) Take the square root of the last term of the given trinomial and put it in the 2nd position in each binomial (4) The signs of each binomial should be the same as the middle term of the given trinomial (1) Find a convenient point in the polynomial to partition (or group) Factoring by (2) Factor within each group Grouping (3) Factor out the Greatest Common Factor across the groups To factor trinomials in the form ax2 + bx + c: Factoring (1) Multiply the a term by the c term Trinomials with a (2) Find the factors of (ac) which will add to the b term Leading (3) Rewrite the b term as the sum of two x terms with coefficients being the Coefficient factors of (ac) Greater Than One (4) Group the first two terms and last two terms each in a set of parentheses (5) Factor out the Greatest Common Factor from each group 15 College Algebra with Trig Name:__________________________________ Review- Rational Expressions Test Date:___________________________________ ANSWER EACH ON A SEPARATE SHEET OF PAPER. SHOW ALL WORK! Factor completely. If the polynomial cannot be factored, write simplified. (1) 6c2 + 13c + 6 (4) y4 – z2 (7) 3d2 – 3d – 5 (2) a2b2 + ab – 6 (5) x5 + 27x2 (8) 72 – 26y + 2y2 (3) t2 – 2t + 35 (6) x4 – 81 (9) x3 + 7x2 + 2x + 14 Perform the indicated operation and simplify your answer. 6a 2 2a 9a 2 1 7 4 x (10) (14) 1 9a 2 6a 1 6a 2 a3 2a (18) 3 x2 3 3 t 2 6t 9 t 2 t 20 1 1 (11) 2 (15) t 2 10 t 25 t 7t 12 x 1 x 7 1 y2 x4 3x 12 2m 18 (19) (12) (16) 3 m9 9m 1 2x 7 x 3 2 5x 2 45 y2 x 2 3x x 2 5x 6 3 3 (13) 4 2x 2 x 6 x2 4 xy xy 1 (17) 6 (20) x 1 24 x y2 2 x 1 x 1 Answer the following word problems, showing all Solve each of the following equations and check: work to explain your answer: 4 5 3x (21) The area of a rectangle is (x2 – x – 6) square (23) x 1 2x 2 4 meters. The length and width are each increased by 9 meters. Write the area of the new rectangle as a trinomial in terms of x. 2 1 1 2a (24) 2 (22) The freshman and sophomore classes both a4 a 2 a 2a 8 participated in a fundraiser. The freshman class collected (4x2 – 1) and the sophomore class collected (6x2 + 7x + 2). Express, in simplest form, the ratio of the sophomore’s collection to the freshman’s collection. 16 FUNCTIONS 17 College Algebra with Trig Name:____________________________________ Lesson- Relations and functions Date:_____________________________________ Objective: To know the definitions of relations and functions. To understand the difference between what is a function and what is not. To be able to determine whether a relation is a function. Definitions: Relation- Function- Domain- Range- PBLM SET. 1. State whether the relation is a function or not: Identify the Domain and Range. a. {(-2, 0), (3, 2), (4, 5)} b. {(6, -2), (3, 4), (6, -6), (-3, 0)} 2. Which relation is a function? Why? (a) (b) (c) (d) 3. Find the Domain and Range of each choice in exercise #2. 4. Determine whether each of the following is a function. Justify your answer. Find the Domain and Range of each. a. f ( x) x3 b. f(x) - x 2 2x - 27 18 19 20 College Algebra with Trig Name:____________________________________ Lesson- Linear Functions Date:_____________________________________ Objectives: To know the various properties of a linear function. To understand the processes for writing and graphing various types of linear functions. Do Now: State the four different types of slope and give an example for each: Linear Function: Forms of Linear Functions: 1. slope-intercept form: 2. standard form: 3. point-slope form Ex 1: Write the linear equation in slope intercept, standard, and point-slope form given that the line passes through (5, 2) and (7, 9) 21 Ex 2: Write the equation of the horizontal line that passes through (-9, 2) Parallel & Perpendicular Linear Function Rules: Parallel Perpendicular Ex 3: Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to 4 the graph of y 3x 5 2 4 Ex 4: Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of y x 3 7 Intercepts x-intercepts: y-intercepts: 1 Ex 5: Find the x- and y- intercepts of f ( x) x 2 . Graph the linear function. 3 22 College Algebra with Trig Name:____________________________________ Lesson- Evaluating Functions Date:_____________________________________ Objectives: To know what it means to evaluate a function. To understand how (and be able) to evaluate a function algebraically and graphically. Notation for a function: What does “evaluate a function” mean? Evaluating Functions Algebraically 1. find f(-1) if f(x) = x2 – 1 2. find h(3) if h(x) = 3x2 3. find f(-7) if f(w) = 16 + 3w – w2 4. find g(m) if g(x) = 2x6 – 10x4 – x2 5. find k(w + 2) if k(x) = 3x + 4 6. find h(a – 2) if h(x) = 2x2 – x + +5 3 Evaluating Functions Graphically 1. 2. 3. 23 College Algebra with Trig Name:____________________________________ Lesson- Graphing Absolute Value Functions Date:_____________________________________ Objective: To learn how to graph a piecewise and absolute value function Do Now: x2 1 State the domain for f ( x) x __________________________________________________________________________________________ Absolute Value Functions y Graph the Following f ( x) x x Now graph each of the following and discuss how each relates to f (x) from above. g ( x) 2 x h( x) 2 x 3 i( x) 2 x 3 y y y x x x 24 College Algebra with Trig Name:____________________________________ Solving Absolute Value Equations Date:_____________________________________ Objectives: To learn to solve absolute value equations and absolute inequalities. Absolute Value Equations ax b c To solve ax b c create 2 equations and solve each. ax b c Example: 3x 1 2 Practice: a. x 1 4 b. 3 y 5 c. 2 3d 4 d . 2m 1 2 25 College Algebra with Trig Name:____________________________________ Lesson- Solving Absolute Value Inequalities Date:_____________________________________ Objectives: To learn to solve absolute value inequalities. Absolute Value Inequalities There are three absolute value situations: Case 1 Case 2 Case 3 ax b c ax b c ax b c ax b c c ax b c Either ax b c or ax b c ax b c Examples: a. 3x 1 2 b. 3x 1 2 c. 3x 1 2 d. 3x 1 2 Practice: a. x 1 4 b. 3 y 5 c. 2 3d 4 d . 2m 1 2 26 27 College Algebra with Trig Name:____________________________________ WKST- Mixed equation/inequality and absolute value set Date:_____________________________________ Answer each of the following neatly and completely in the space provided. Solve and graph each inequality: 1. x 7x 6 2. x 3 3(2 x 1) 3. x3 4 4. 2x 5 x 1 5. 2x 3 5 6. 2 x 8 7. x 2 2 x 24 0 8. x 2 10 x 1 0 28 9. x2 4 0 10. x 2 8x 7 0 11. 3x 2 10x 8 12. 5d 7 28 13. Explain why the solution set of 14. Explain why the solution set of x 2 9 0 is all real numbers. x 2 16 0 is empty. 29 College Algebra with Trig Name:____________________________________ Lesson- Graphing Piecewise Functions Date:_____________________________________ Objective: To learn how to graph piecewise functions. Do Now: Graph: f ( x) 3x 2 for 3 x 0 y x What is a piecewise function? Graph the following: y 2 x if x 0 f ( x) 2 if x 0 x 2 x 1 if x 0 g ( x) 2 y x if x 0 x 30 31 College Algebra with Trig Name:__________________________________ Mixed Wkst: Graphing Absolute and Piecewise Functions Date:___________________________________ y 1. Graph the following function: x 2 if x 1 f ( x) x 2 if x 1 x 2. Graph each function, and state the domain and range (1) f ( x) x 3 (2) f ( x) 3 x 2 y y x x 32 College Algebra with Trig Name:__________________________________ Lesson- One-to-One and Onto Date:___________________________________ Objectives: To know what it means for a function to be One-to-one or Onto. To be able to distinguish between One-to-one and Onto. Definitions Abscissa- Ordinate- One-to-one Functions A function is one-to-one when no two ordered pairs in the function have the same ordinate and different abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test. If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an inverse**) Onto Functions A function is Onto if each ordinate associated with an abscissa. Multiple abscissas may map onto the same ordinate. (**Note: if a function does not use all y-values in a Cartesian plane, it cannot be onto) 33 Examples: Determine whether the following refers to a function one-to-one, onto, both or neither. Explain your reasoning. 1) f ( x) 2x 1 2) f ( x) x 2 1 3) f ( x) 3 5 x 4) 5) 6) 7) 8) 9) 34 College Algebra with Trig Name:__________________________________ Lesson- Composition & Inverse of Functions Date:___________________________________ Objective: To know how to find the composition and inverse of a function. To understand the process for finding the composition and inverse of a function. To be able to recognize an inverse graphically. Do Now: Evaluate f ( x) x 3 x for x 2 Composition of Functions “following” one function with another. Notation: Both of the following mean “f following g.” f ( g ( x)) and ( f g )(x) Ex 1: f ( x) x 5 g ( x) 4 x Find: a) f ( g ( x)) b) f (g (2)) c) ( g f )(3) d) g ( f ( x)) Would you say that a composition is a commutative operation? Why/why not? h( x ) x 2 Ex 2: Find: a) h(r ( x)) b) r (h( x)) c) h(r (5)) r ( x) x 3 35 Inverse Functions Definition: Steps: 1. Write the equation in terms of x and y. 2. Switch the x with the y. 3. Solve for y. Ex 1: Find the inverse of y 4 x 8 Ex 2: Find the inverse of f ( x) 5x 2 Ex 3: Find the inverse of g ( x) x 2 4 Ex 4: Graph y 4 x 8 and it’s inverse on the axes below. y x Ex 5: Looking at the graph of a line, can you find a way to graph it’s inverse? 36 College Algebra with Trig Name:__________________________________ Lesson- Operations with Functions Date:___________________________________ Operations with Functions: given functions f and g sum: f g( x) f ( x) g( x) difference: f g( x) f ( x) g( x) product: f g( x) f ( x) g( x) f f ( x) quotient: ( x) g , where g( x) 0 g( x) Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each (1) f ( x) 3x 1; g ( x) x (2) f ( x) 5 x 4 ; g ( x) x 2 1 (3) f ( x) 5 x ; g ( x) x 1 37 College Algebra with Trig Name:__________________________________ Lesson- Function transformations Date:___________________________________ Objectives: To know the rules for various transformations such as: translations, reflections, symmetry, rotations, and dilations. To understand the process for transforming coordinates, lines, and curves. To be able to conduct various transformations and compositions of transformations. Do Now: Sketch the graph the following polynomial: f ( x) x 1 y x Definitions: Pre-Image: Image: Types of Transformations and their specific rules 38 Extra Space 39 Unit 2: Relations & Functions Definitions, Properties & Formulas Relation a set of ordered pairs (x, y) Domain the set of all x-values of the ordered pairs Range the set of all y-values of the ordered pairs a relation in which each element of the domain is paired with exactly one element Function in the range. the slope, m, of the line through (x1, y1) and (x2, y2) is given by the following Slope y y1 equation, if x1 x2: m 2 x 2 x1 y y y y Types of Slope x x x x Positive Negative Zero Undefined horizontal line: vertical line: x y=b =a y-intercept where the graph crosses the y-axis x-intercept where the graph crosses the x-axis y = mx + b Slope-Intercept Form where m represents the slope and b represents the y-intercept of the linear equation Ax + By = C Standard Form where A, B, and C are constants and A 0 (positive, whole number) y – y1 = m(x – x1) Point-Slope Form where m represents the slope and (x1, y1) are the coordinates of a point on the line of the linear equation Two non-vertical lines in a plane are parallel if and only if their slopes are equal Parallel Lines and they have no points in common. (Two vertical lines are always parallel.) Perpendicular Two non-vertical lines in a plane are perpendicular if and only if their slopes are Lines negative reciprocals. (A horizontal and a vertical line are always perpendicular.) 40 Vertical Line Test If any vertical line passes through two or more points on the graph of a relation, (VLT) then it does not define a function. Horizontal Line If any horizontal line passes through two or more points on the graph of a relation, Test (HLT) then its inverse does not define a function. One-to-One a function where each range element has a unique domain element Functions (use HLT to determine) Onto Functions All values of y are accounted for Inverse Relations f -1(x) is the inverse of f(x), but f -1(x) may not be a function & Functions (use HLT to determine) To find f -1(x): (1) let f(x) = y Writing Inverse (2) switch the x and y variables Functions (3) solve for y (4) let y = f -1(x) sum: (f + g)(x) = f(x) + g(x) difference: (f – g)(x) = f(x) – g(x) Operations with product: (f g)(x) = f(x) g(x) Functions f f ( x) quotient: ( x) g , where g( x) 0 g( x) Reflections: rx axis ( x, y ) ( x, y ) ry axis ( x, y ) ( x, y ) rorigin ( x, y ) ( x, y ) Dilations: Dk ( x, y ) (kx, ky) Transformations Translations: Ta ,b ( x, y ) ( x a, y b) Rotations: R0,90 ( x, y ) ( y, x) 41 College Algebra with Trig Name:__________________________________ Review- Function test Date:___________________________________ Objective: To review the material that you will be tested on as part of Test #1-Functions. These topics are in the outline below: Functions a. Identifying functions b. Domain and Range of functions c. Linear Function i. Finding x and y intercepts ii. Writing and graphing the equation of line in slope intercept form iii. Parallel and perpendicular lines and their graphs d. Evaluating functions graphically e. Evaluating functions algebraically f. Absolute Value Functions g. Piecewise Functions h. Identifying one-to-one functions i. Identifying onto functions j. Composition of functions k. Inverse functions l. Operations with Functions m. Transformation of functions Below you will find a sample of the types of problems you can expect to see on the test. a. Which graph of a relation is also a function? (a) (b) (c) (d) b. Determine the Domain and Range of: i. f ( x) 3 x 4 ii. g ( x) x 2 9 ci. Find the x and y intercepts for the following linear equations: 1. x 3 y 7 2. 3x 4 y 12 42 cii. Write and graph the equation of the line given the following information: 1. m 3, and passes through (3,2) 2. passes through (5,1) and (2,0) ciii. 1. Write & graph the equation of the line that is parallel to y 3x 2 and passes through (4,1). 2. Write and graph the equation of the line that is perpendicular to y 3x 2 and passes through its x intercept. d. If the following graph is y = f(x), what is the value of f(1)? (a) -1 (b) -2 (c) 1 (d) 2 43 e. Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1) f. What are the significance of the a,h,k values in the standard form of an absolute value function? Write f ( x) 2 x 2 2 2 g. h. Which function is not one to one? (a) (b) (c) (d) i. Which function is not onto? (a) (b) (c) (d) j. Given f ( x) 3x 4; g ( x) x 2 9 , find ( f g )(x) and ( g f )(x) . 44 k. Find the inverse of the following and state the domain. 4 a. f(x) = 5x + 2 b. f ( x) x3 l. Perform the four basic operations on f ( x) 3x 4; g ( x) x 2 9 and determine the domain of the result. m. Complete the following transformations on graph paper. Label your images. a. rx axis (2,1) b. D 2 [ f ( x) x 2 1] c. R0,90 [ g ( x) 2 x 1] d. T2,3 [h( x) 2x 2 4x 2] --------------------------------------------------------------------------------------------------------------------------------------- a. b. y y x x c. y d. y x x 45 QUADRATICS 46 College Algebra with Trig Name:____________________________________ Rational Exponents Review Date:_____________________________________ Objective: To learn to use the rules of exponents, including zero, negative and fractional exponents, in multiplication and division of monomials. Zero Exponents: Any value raised to the 0th power is equal to 1. Example 1: (5 x 2 ) 0 ? Negative Exponents: When a monomial is raised to a negative exponent, take the reciprocal of the base to get rid of the negative in the exponent. 3 2x Example 2: ? 3 Fractional Exponents: When a monomial is raised to a fractional exponent, the numerator represents the power the monomial should be raised to. The denominator represents the root of the monomial that should be taken. 3 16 2 Example 3: ? 25 2 125 3 Example 4: =? 64 Multiplying: If two monomials are being multiplied, you generally: Multiply the coefficients Keep the variable and add the exponents. Example 5: (3 x 2 )( 2 x 3 ) ? 47 But sometimes you are asked to multiply in a different way. Example 6: (3 x 3 y ) 3 ? What is another way that you can write Example 2? What is a general rule that you can use to simplify this process? Dividing: If two monomials are being divided, you generally: Divide (or simplify) the coefficients Keep the variable and subtract (or cancel) the exponents. 6x 2 y Example 3: 3 3x y 2 2 More Practice Exercises Simplify the following. 1) (2 x 3 y 0 )(3 x 1 y ) 2 2) (2ab 3 )(2a 1b 2 ) 2 m2n4 3) m 2 n 1 25 2 4) 49 48 Some More Complicated Practice Exercises Simplify each of the following. 2 7 1 x y 6 3 3 49 x y 3 10 4 12 a b 3 5) 3 4 x y 6) 1 7) 1 7 x y 12 3 5c 2 Practice: Simplify 1 25 2 12a 4 b 3 1. 2. 49 5c 1 27 3 3. 4. 49 x 8 y 10 64 3 xy 4 z 3 x 2 5. 2 5 x yz 6. Challenge: x 2 49 College Algebra with Trig Name:____________________________________ Lesson- Graphing Quadratic Functions Date:_____________________________________ Objective: To graph a quadratic function using the Roots Axis of symmetry Vertex y Do Now: Graph the following function. f ( x) 3 x 1 x Roots: Axis of symmetry: Vertex: __________________________________________________________________________________________ Graph each of the following on the same set of axes by finding the intercepts, axis of symmetry and vertex. 1. f ( x) x 2 4 x 2. g ( x) x 2 2 x 5 3. h ( x ) x 2 4 x 4 Based on the graphs above, is there a shortcut for determining if the parabola opens up or down? 50 51 College Algebra with Trig Name:____________________________________ Challenge Problem Set- Quadratics and their graphs Date:_____________________________________ 1. Given a quadratic function: y ax 2 bx c , determine the value of “b” if the vertex is ( 2, 2) and the y- intercept is (0, -2). 2. Given a quadratic function: y ax 2 bx c , determine the value of “b” if the vertex is ( p, p) and the y- intercept is (0, -p). Hint: Look at your solution to #1. 52 College Algebra with Trig Name:____________________________________ Lesson- Zero Product Rule Date:_____________________________________ Objective: To apply the zero product method for finding the roots of a quadratic function. Do Now: Create a table of values to graph the following function: h( x) x 2 14 x 1 Zero product rule: Find the roots of the given quadratic using the zero product method: 1. x 2 4 x 3 0 2. x 2 7 x 6 0 3. x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 2 x 3 3x x 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 2 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 53 College Algebra with Trig Name:____________________________________ Lesson- Completing the square Date:_____________________________________ Objective: To find the roots of a quadratic function by completing the square. Do Now: Find the roots of the quadratic function by using the zero-product rule. x 2 25 200 Completing the square: Find the roots of each of the following by completing the square (any imaginary answers should be put in simplest a bi form). 1. x 2 4 x 3 0 2. x 2 7 x 5 0 3. 2 x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 x 3 2x x 2 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 3 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 54 College Algebra with Trig Name:____________________________________ Lesson- Quadratic Formula Date:_____________________________________ Objective: To find the roots of a quadratic function by using the quadratic formula Do Now: Find the roots of the quadratic function by using the completing the square method. x3 8 2 x3 Quadratic Formula Derivation: Find the roots of each of the following by using the quadratic formula (any imaginary answers should be put in simplest a bi form) 1. x 2 4 x 3 0 2. x 2 7 x 5 0 3. 2 x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 2 x 3 2x x 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 3 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 55 College Algebra with Trig Name:____________________________________ Lesson- Quadratic Formula Workspace Date:_____________________________________ Complete #1-15 in the space below 56 College Algebra with Trig Name:____________________________________ Lesson- Standard form of a quadratic function Date:_____________________________________ Objective: To write and graph a quadratic function in standard form Do Now: Find the roots of the quadratic function by using the quadratic formula. x3 8 2 x2 Standard form of a Quadratic Function: f ( x ) a ( x h) 2 k where (h,k) is the vertex and a determines whether that vertex is a maximum or minimum. Writing a quadratic in standard form: Example: x 2 4 x 7 __________________________________________________________________________________________ Write each of the following in standard form and determine the vertex and whether that vertex is a maximum or minimum. 1. x 2 4 x 3 0 2. x 2 7 x 5 0 3. 2 x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 2 x 3 2x x 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 3 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 57 College Algebra with Trig Name:____________________________________ Lesson- Standard form of a quadratic function Workspace Date:_____________________________________ Complete #1-15 in the space below 58 College Algebra with Trig Name:____________________________________ HW- Solving for the Roots of Quadratics & Standard Form Date:_____________________________________ Answer each of the following neatly and completely. Write the following equations in standard form, state the values of a & (h, k). 1. 5x 2 2 x 4 2. - x 2 4 2 x 3. 5x 4 6 x 2 Find the roots for each of the following equations using any of the 3 methods (if possible) 4. z 2 5 z 4 0 5. x 2 11x 24 0 6. s 2 s 0 7. 2 x 2 5 x 2 0 8. x 2 81 9. y 2 6 y 1 2 7 10. x x 1 2 6 59 College Algebra with Trig Name:____________________________________ Lesson- Describing the nature of quadratic roots Date:_____________________________________ Objective: To use the discriminant to determine the nature of the roots of quadratics Do Now: Write the following quadratic in standard form x3 8 2 x2 Discriminant: Determine the nature of the roots of each of the following. In each case, determine which method of finding the roots of quadratics would be the best to use? 1. x 2 4 x 3 0 2. x 2 7 x 5 0 3. 2 x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 x 3 2x x 2 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 3 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 60 College Algebra with Trig Name:____________________________________ Lesson- Sum and product of roots Date:_____________________________________ Objective: To find the sum and product of the roots of a quadratic. To find the standard form equation of a quadratic given roots or sum and product of roots To find shortcuts for different transformations of quadratics Do Now: Change f ( x) x 2 4 x 3 into standard form and identify the vertex Sum and Product formulas: Proof: Writing quadaratic equations given roots Procedure: Examples: Write the standard form equation of the quadratic given the following roots: 1. x= 1,7 2. x= 4i, -4i 3. x= e-fi, e+fi Writing quadaratic equations given the sum and product of the roots Procedure: Examples: Write the standard form equation of the quadratic given the following sum and product of roots. 1. sum=12, product= 16 2. sum= -7, product= 10 61 College Algebra with Trig Name:____________________________________ Lesson: Quadratic Word Problems Date:_____________________________________ Objective: To learn to interpret quadratic word problems, write symbolically and solve. x x2 Do Now: Solve for x: x 1 2 __________________________________________________________________________________________ Method: __________________________________________________________________________________________ 1. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. When does the object strike the ground? 2 A picture has a height that is 4/3 its width. It is to be enlarged to have an area of 192 square inches. What will be the dimensions of the enlargement? 3 You have to make a square-bottomed, unlidded box with a height of three inches and a volume of approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from each corner, scoring between the corners, and folding up the edges. What should be the dimensions of the cardboard, to the nearest quarter inch? 4 A factory produces lemon-scented widgets. You know that each unit is cheaper, the more you produce, but you also know that costs will eventually go up if you make too many widgets, due to storage requirements. The guy in accounting says that your cost for producing x thousands of units a day can be approximated by the formula C = 0.04x2 – 8.504x + 25302. Find the daily production level that will minimize your costs. 62 College Algebra with Trig Name:____________________________________ HW: Quadratic Word Problems Date:_____________________________________ Answer each of the following neatly and completely on separate graph paper. Be sure to include graphs for each problem. 1. An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be the object's maximum height? When will it attain this height? 2. The product of two consecutive negative integers is 1122. What are the numbers? 3. You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area? 63 College Algebra with Trig Name:________________________________________ Wkst- Quadratic Word Problems Date:_____________________________________ A rectangular dog pen is to be made along an existing fence. The total length of the new fence Existing fence is to be 100 feet and the width is x. x pen x a) Express, in simplest from, the area of the pen as a function of x. b) What is the width that gives the maximum area? Show or explain how you arrived at your answer. c) What is the maximum area? 64 Another version of the fence problem… The owner takes Existing wall down the fence and now wants to use the 100 feet of fencing to make two adjacent dog pens against the existing wall. See x x x the diagram at the right. a) If x is the length of fence perpendicular to the existing wall, express, as a function of x, the length of the fence parallel to the wall. b) Express, as a function of x, the total area of both pens. c) What is the value of x (to the nearest tenth) that gives the maximum area? Show or explain how you arrived at your answer. d) What is the maximum total area(to the nearest whole unit)? 65 College Algebra with Trig Name:____________________________________ Lesson- Solving quadratic linear systems Date:_____________________________________ Objective: To learn how to solve systems of equations involving a line and a parabola. Do Now: Solve for x: 2 x 2 3x 9 0 __________________________________________________________________________________________ Solving a quadratic linear system Graphical Process: Algebraic Process: Examples: y x 2 2x 3 y x 2 2x 1 y x2 1 1. 2. 3. 2x y 2 y 2x 3 y x 1 66 College Algebra with Trig Name:____________________________________ Lesson- Quadratic transformations Date:_____________________________________ Objective: To find shortcuts for different transformations of quadratics Do Now: Write the standard form equation given: f ( x) 2 x 2 4 x 3 Parent graph: f ( x) 2( x 3) 2 5 Reflections: 67 Translations: Dilations: 68 Shortcut Rules for Quadratic Transformations Type f ( x ) a ( x h) 2 k Reflection over the x axis Negate a,k Reflection over the y axis Negate h Reflection over the origin Negate a,h,k Tm , n f ( x ) a ( x h m) 2 k n Dn Divide a by n and multiply (h,k) by n Problems: For each of the following: a. reflect over the x-axis, y-axis, origin and line y = x. b. translate (4, 2) c. dilate by a factor of 3 d. dilate by a factor of 1/3 e. graph each on separate graph paper. 1. y x2 2. f ( x) ( x 1) 2 3 69 College Algebra with Trig Name:____________________________________ Lesson- Quadratic Inequalities Date:_____________________________________ Objective: To solve and graph quadratic inequalities Do Now: Describe the nature of the roots of the given quadratic. x3 8 2 x Let a = 1st factor and b=2nd factor If ab=0 ab>0 ab<0 Then a=0 a<0 and b<0 a<0 and b>0 Or b=0 a>0 and b>0 a>0 and b<0 Example: Solve and graph the solution set for each: a. x 2 16 0 b. x 2 7 x 0 c. 2 x 2 11x 5 0 d . x 2 8 x 20 70 College Algebra with Trig Name:____________________________________ WKST- Quadratic Inequalities Date:_____________________________________ Solve and sketch the graph of the solution set for each: 1. x 2 4 x 3 0 2. x 2 7 x 6 0 3. x 2 10 x 0 4. 3x 2 x 5. x( x 4) 5 x 3 3x x 2 6. x 2 2( x 12) 7. 8. 9. x 2 25 0 10. 16 x 2 64 2 x 1 2 4 x 24 13 11. 9 x 2 6 x 1 0 12. 6 x 2 x 2 0 13. 14. x 6 15. 2 x 2 7 x 4 0 6 x x6 71 College Algebra with Trig Name:____________________________________ Lesson/Activity- Quadratic Exploration Date:_____________________________________ 1. f ( x) x 2 6 x 9 a. Use the Quadratic Transformer to determine the vertex h, f (h) of f ( x) x 2 6 x 9 . b. Find the following (where “h” is the “x” value of the coordinates of the vertex) i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 2. f ( x) x 2 5 x 6 a. Use the Quadratic Transformer to determine the vertex h, f (h) of f ( x) x 2 5 x 6 . b. Find the following (where “h” is the “x” value of the coordinates of the vertex) i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 3. f ( x) x 2 4 a. Use the Quadratic Transformer to determine the vertex h, f (h) of f ( x) x 2 4 . b. Find the following (where “h” is the “x” value of the coordinates of the vertex) i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 72 4. By now you should be noticing a pattern in the differences found in part c of the preceding questions. 5. f (x) 2x2 8 a. Use the Quadratic Transformer to determine the vertex h, f (h) of f (x) 2x2 8 . b. Find the following (where “h” is the “x” value of the coordinates of the vertex) i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 6. f ( x) 3 x 2 6 x 5 a. Use the Quadratic Transformer to determine the vertex h, f (h) of f ( x) 3x 2 6 x 5 . b. Find the following (where “h” is the “x” value of the coordinates of the vertex) i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 73 7. At first glance, it seems that there is no discernible pattern of consistency between part c of #5,6. But closer inspection leads to an interesting discovery. If you are not seeing it yet, try a quadratic function in which the leading coefficient is 4, 5, 6… 8. Let’s prove the relationship: a. Determine the axis of symmetry of: f ( x) ax 2 bx c and call it “h.” (Hint: You should already know this!) b. Find: i. f (h) ii. f (h 1) iii. f (h 2) iv. f (h 3) c. Find the difference between: i. f (h 3) & f (h) ii. f (h 2) & f (h) iii. f (h 1) & f (h) 74 EXPONENTIAL AND LOG FUNCTIONS 75 College Algebra with Trig Name:__________________________________ Lesson- properties, equations with exponents and power and exponential functions Date:___________________________________ Objectives: use the properties of exponents solve equations containing rational exponents examine power and exponential functions Do Now: Use the exponential properties to simplify and rewrite the following expressions: (1) ax ay (2) a x y (3) ab x x a (4) b ax (5) ay (6) a x (7) a0 __________________________________________________________________________________________ In Small Groups: Use each example in the “Do Now” to arrive at general rules as they apply to monomials with exponents. Using Exponential Function Properties to Solve for x: Process 1 Process 2 Examples (each relates to “Process 1”): 44 x1 42 x2 45 x1 162 x1 3x 9 x4 2 1. 2. 3. 76 More Examples (each relates to “Process 2”): 4. x 4 81 5. x 1 4 6. (2 x 1)5 32 Power function: exponential function: Small Group Activity On your graphing calculator, simultaneously graph: y = 0.5x, y = 0.75x, y = 2x, y = 5x (1) What is the range of each exponential function? (2) What is the behavior of each graph? (3) Do the graphs have any asymptotes? (4) (a) What point is on the graph of each function? (b) Why? Characteristics of graphs of y = nx n>1 0<n<1 domain range y-intercept behavior horizontal asymptote vertical asymptote Extension: Graph the exponential functions y = 2x, y = 2x + 3, and y = 2x – 2 on the same set of axes. Compare and contrast the graphs using a table similar to the one above. 77 College Algebra with Trig Name:__________________________________ Lesson- Graphing exponential functions, exponential growth and decay Date:___________________________________ Objectives: graph exponential functions use exponential functions to determine growth and decay Using Exponential Functions for Real World Applications: Exponential growth: Exponential decay: Exponential Growth or Decay: N = N0 (1 + r)t (1) Write a formula that represents the average growth of the population of a city with a rate of 7.5% per year. Let x represent the number of years, y represent the most recent total population of the city, and A is the city’s population now. What is the expected population in 10 years if the city’s population now is 22,750 people? Graph the function for 0 x 20. 78 __________________________________________________________________________________________ (2) Suppose the value of a computer depreciates at a rate of 25% a year. Determine the value of a laptop computer two years after it has been purchased for $3,750. (3) Mexico has a population of about 100 million people, and it is estimated that the population will double in 21 years. If population growth continues at the same rate, what will be the population in: (a) 15 years (b) 30 years (c) graph the population growth for 0 time 50 __________________________________________________________________________________________ (4) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a rate of 14% per week. What is the expected population in 22 weeks? 79 (5) In 1990, Exponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%. Find the projected population for 2010. (6) Find the projected population of each location in 2015: (a) In Honolulu, Hawaii, the population was 836,231 in 1990. The average yearly rate of growth is 0.7%. (b) The population in Kings County, New York has demonstrated an average decrease of 0.45% over several years. The population in 1997 was 2,240,384. 80 College Algebra with Trig Name:____________________________________ Lesson- More exponential function graphs, Population growth, half-life Objectives: graph exponential functions use exponential functions to determine population growth and half-life decay (1) The population of Los Angeles County was 9,145,219 in 1997. If the average growth rate is 0.45%, predict the population in 2010. Graph the equation for 0 time 20. (2) Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a half-life of 2.67 days. If the initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be left over after: (a) ½ day (b) 1 week 81 (3) If a farmer uses 25 pounds of insecticide, assuming its half-life is 12 years, how many pounds (rounded to the nearest tenth) will still be active after: (a) 5 years (b) 20 years (4) In 2000, the chicken population on a farm was 10,000. The number of chickens increased at a rate of 9% per year. Predict the population in 2005. Graph the equation for 0 time 15. (5) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth continues at the same rate, find the population (rounded to the nearest million) in: (a) 10 years (b) 30 years 82 College Algebra with Trig Name:__________________________________ Lesson- Compound Interest Date:___________________________________ Objectives: use exponential functions to determine compound interest Do Now: (1) A laser printer was purchased for $300 in 2001. If its value depreciates at a rate of 30% a year, determine how much it will be worth in 2007. (2) Rates can be compounded in different increments per year. Exponential growth occurs how often if the rate is compounded: annually: bi-annually: quarterly: monthly: weekly: daily: The general equation for exponential growth is modified for finding the balance in an account that earns compound interest. nt r Compound Interest: A P1 n 83 __________________________________________________________________________________________ (1) If Charlie invested $1,000 in an account paying 10% compounded monthly, how much will be in the account at the end of 10 years? (2) Mike would like to have $20,000 cash for a new car 5 years from now. How much should be placed in an account now if the account pays 9.75% compounded weekly? (3) Suppose $2,500 is invested at 7% compounded quarterly. How much money will be in the account in: (c) ¾ year (d) 15 years __________________________________________________________________________________________ (4) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in: (e) ½ year (f) 10 years 84 (5) How much money must Cindy invest for a new yacht if she wants to have $50,000 in her account that earns 5% compounded quarterly after 15 years? (6) Carol won $5,000 in a raffle. She would like to invest her winnings in a money market account that provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in the account at the end of 10 years? Show your work and explain your answer. 85 College Algebra with Trig Name:__________________________________ Lesson: Exponential Functions with base e Date:___________________________________ Objective: use exponential functions with base e Euler Savings Bank provides a savings account that earns compounded interest at a rate of 100%. You may choose how often to compound the interest, but you can only invest $1 over the course of one year. 86 Exponential Growth or Decay (in terms of e): N = N0 ekt (1) According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume that the initial temperature Ti is 90F and that k = 0.275. (a) Write a function to model the rate at which the liquid cools. (b) Find the temperature T of the liquid after 4 minutes (t) (c) Graph the function and use the graph to verify your answer in part (b) 87 (2) Suppose a certain type of bacteria reproduces according to the model B = 100 e0.271 t , where t is the time in hours. (a) At what percentage rate does this type of bacteria reproduce? (b) What was the initial number of bacteria? (c) Find the number of bacteria (rounded to the nearest whole number) after: (i) 5 hours (ii) 1 day (iii) 3 days (3) A city’s population can be modeled by the equation y = 33,430e0.0397 t , where t is the number of years since 1950. (a) Has the city experienced a growth or decline in population? (b) What was the population in 1950? (c) Find the projected population in 2010. 88 College Algebra with Trig Name:__________________________________ HW- Compound Interest Date:___________________________________ (1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in the account at the end of: (a) 6 years 3 months (b) 204 months (2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in the account at the end of: (a) 5.5 years (b) 12 years 89 __________________________________________________________________________________________ (3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to pay for the note now if the note gains value at a rate of 9% compounded continuously? (4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously. She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be enough funds in Niki’s account in 6 years to meet her goal? Explain your answer. 90 College Algebra with Trig Name:__________________________________ Lesson- Continuous Compound Interest Date:___________________________________ Objective: use exponential functions to determine continuously compounded interest Continuously Compounded Interest: A = Pert (1) Tim and Kerry are saving for their daughter’s college education. If they deposit $12,000 in an account bearing 6.4% interest compounded continuously, how much will be in the account when she goes to college in 12 years? (2) Paul invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously. If Paul made the investment on January 1, 1995, and the account was worth $12,000 on January 1, 1999, what was the original amount in the account? 91 (3) Compare the balance after 30 years of a $15,000 investment earning 12% interest compounded continuously to the same investment compounded quarterly. (4) Given the original principal, the annual interest rate, the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment: (a) P = $1,250; r = 8.5%; t = 3 years; compounded semi-annually (b) P = $2,575; r = 6.25%; t = 5 years 3 months; compounded continuously 92 College Algebra with Trig Name:__________________________________ HW- Compound Interest Date:___________________________________ (1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in the account at the end of: (a) 6 years 3 months (b) 204 months (2) If you invest $7,500 in an account paying 8.35% compounded continuously, how much money will be in the account at the end of: (a) 5.5 years (b) 12 years 93 __________________________________________________________________________________________ (3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to pay for the note now if the note gains value at a rate of 9% compounded continuously? (4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously. She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be enough funds in Niki’s account in 6 years to meet her goal? Explain your answer. 94 College Algebra with Trig Name:____________________________________ Lesson- Properties of a logs, rewriting Exponential functions as logarithms, log graphs Date:_____________________________________ Objective: To learn what a logarithm is To learn the properties of logs To learn to rewrite an exponential function as a logarithm Graphing logs Do Now: Solve for x: 3x 9 x1 and check. _________________________________________________________________________________________ What is a logarithm? Logarithms are inverses of exponential functions. Logarithms are functions because exponential functions are one-to-one functions. We cannot solve an equation like: y 2 x using the algebraic techniques we have learned so far. Therefore, we must try an alternative technique. Rule: x b y is equivalent to y log b x The log to the base b is the exponent to which b must be raised to obtain x. Properties of Logs logb 1 0 logb b 1 logb b x x blog b x x , where x > 0 log b MN log b M log b N M logb logb M logb N N log b Mp p log b M Example: Convert each into logarithmic form Convert each into logarithmic form 1 1. y 2 x 4. log 25 5 2 2. 3 9 5. log a b c 1 1 3. 51 6. log 3 2 5 9 95 _________________________________________________________________________________________ What is a Natural Logarithm? Rule: x b y is equivalent to y log b x The log to the base b is the exponent to which b must be raised to obtain x. Properties of Logs ln 1 0 ln b 1 ln e x x eln x x , where x > 0 ln MN ln M ln N M ln ln M ln N N ln M p p ln M Example: Convert each into logarithmic form Convert each into logarithmic form 1. y e x 4. ln 5 x 2. e x 5. ln b c 1 3. e 1 6. ln y 2 e Example: Graph each of the following on the same set of axes using the graphing calculator. y 1. y 2 x 2. x 2 y 3. log 2 y x 4. log 2 x y 5. y e x 6. x e y x 7. ln y x 8. ln x y 96 College Algebra with Trig Name:__________________________________ Lesson/HW- Simplify log expressions, common logs, evaluate Date:___________________________________ Objectives: simplify expressions using the properties of logarithmic functions define common logarithms evaluate expressions involving logarithms Problem Set: write the following expressions in simpler logarithmic forms: 1 (1) log b u2 v 7 (2) logb a2 2 m3 u (3) log b 1 (4) logb 2 vw n 3 n (5) logb x (6) logb 2 p q3 (7) Use logarithmic properties to find the value of x (without using a calculator): 1 2 logb x logb 9 logb 8 logb 6 2 3 97 Write each expression in terms of a single logarithm with a coefficient of one: u2 ie : 2 log b u log b v log b v (8) 5 logb x 4 logb y (9) 2 logb x logb y 1 (11) 8 logb c (10) 3 logb x 2 logb y logb z 4 3 1 (12) logb w 2 logb u (13) logb (a2 b3 ) 2 3 Common Logarithm: log 10 log x Change of Base Formula: log a ln a log p a log b a log b ln b log p b Given loga n, evaluate each logarithm to four decimal places: (14) log 8 172 (15) log 6 1.258 (16) log13 0.0065 Extension: Given y = logb n, what can you determine about the log value (y) based on b and n? 98 College Algebra with Trig Name:____________________________________ Lesson/HW- Properties of Logarithmic Functions, Simplifying logarithmic expressions Date:_____________________________________ Objective: examine properties of logarithmic functions simplify expressions using the properties of logarithmic functions Use the properties of logarithmic functions to solve for x: (1) log 5 x 2 (2) log 4 64 x log x 8 3 2 (3) (4) log8 x 3 Use the properties of logarithmic functions to simplify each expression: (5) log8 8 (6) log 0.5 1 (7) , log10 1000 (8) log 2 64 (9) log7 343 (10) log10 0.001 (11) loge e (12) log5 3 5 99 Write the following expressions in simpler logarithmic forms: (13) logb x 6 y 9 v7 (14) logb u8 mn 1 (15) log b (16) logb pq a4 (17) logb 5 x (18) logb 3 x 2 y 2 100 College Algebra with Trig Name:__________________________________ Lesson- Natural Log Word Problems Date:___________________________________ Objectives: solve real-world applications with natural logarithmic functions Do Now: Laura won $2,500 on a game show. She would like to invest her winnings in an account that earns an interest rate of 12% compounded continuously. Does she have to invest all of it in order to have $4,000 in the account at the end of 4 years to put a down payment on a new sailboat? Show your work and explain your answer. (1) Ana is trying to save for a new house. How many years, to the nearest year, will it take Ana to triple the money in her account if it is invested at 7% compounded annually? (2) At what annual percentage rate (to the nearest hundredth of a percent) compounded continuously will $6,000 have to be invested to amount to $11,000 in 8 years. 101 __________________________________________________________________________________________ (3) In 1990, Exponential City had a population of 142,000 people. In what year will the city have a population of about 200,000 people if it was growing at an exponential rate of k = 0.014? (4) If $5,000 is invested at an annual interest rate of 5% compounded quarterly, how long will it take the investment to double? (5) What was the annual interest rate (to the nearest hundredth of a percent) of an account that took 12 years to double if the interest was compounded continuously and no deposits or withdrawals were made during the 12-year period? 102 College Algebra with Trig Name:__________________________________ Lesson- More natural log word problems Date:___________________________________ Objective: solve real-world applications with natural logarithmic functions (1) If a car originally costs $18,000 and the average rate of depreciation is 30%, find the value of the car to the nearest dollar after 6 years. (2) How many years, to the nearest year, will it take for the balance of an account to double if it is gaining 6% interest compounded semiannually? (3) When Rachel was born, her mother invested $5,000 in an account that compounded 4% interest monthly. Determine the value of this investment when Rachel is 25 years old. 103 __________________________________________________________________________________________ (4) The decay of carbon-14 can be described by the formula A A 0 e 0.000124 t . Using this formula, how many years, to the nearest year, will it take for carbon-14 to diminish to 1% of the original amount? (5) In 2002, a farmer had 400 pigs on his farm. He estimated that this population of pigs will double in 15 years. If population growth continues at the same rate, predict the number of pigs in: a. 2010 b. 2030 (6) If the world population is about 6 billion people now and if the population grows continuously at an annual rate of 1.7%, what will the population be (to the nearest billion) in 10 years from now? 104 __________________________________________________________________________________________ (7) If $100 is invested in an account that has an interest of 7% compounded quarterly, how long will it take for the balance to reach a value of $1,000? (8) What interest rate (to the nearest hundredth of a percent) compounded monthly is required for an $8,500 investment to triple in 5 years? (9) An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision. However, even optical instruments have their limitations. The limiting magnitude L of any optical telescope with lens diameter D, in inches, is given by the equation L 8.8 5.1 logD . Use this equation to find the following to the nearest tenth: a. the limiting magnitude for a homemade 6-inch reflecting telescope. b. the diameter of a lens that would have a limiting magnitude of 20.6. 105 Unit 6: Exponential & Logarithmic Functions Definitions, Properties & Formulas Properties of Exponents Property Definition Product x a xb x a b xa Quotient b x a b , where x 0 x Power Raised to a Power (xa)b = xab Product Raised to a Power (xy)a = xa ya a x xa Quotient Raised to a Power a , where y 0 y y Zero Power x0 = 1, where x 0 1 Negative Power x n , where x 0 xn 1 x n x n Rational Exponent for any real number x 0 and any integer n > 1 and when x < 0 and n is odd N = N0 (1 + r)t Exponential where: N is the final amount, N0 is the initial amount, t is the number of time Growth/Decay periods, and r is the average rate of growth(positive) or decay(negative) per time period nt r A P1 Compound n Interest (Periodic) where: A is the final amount, P is the principal investment, r is the annual interest rate, n is the number of times interest is compounded each year, and t is the number of years N = N0 ekt Exponential Growth/Decay where: N is the final amount, N0 is the initial amount, t is the number of time periods, and k (a constant) is the exponential rate of growth(positive) or (in terms of e) decay(negative) per time period Continuously A = Pert Compounded where: A is the final amount, P is the principal investment, r is the annual Interest interest rate, and t is the number of years 106 Logarithmic are inverses of exponential functions Functions a logarithm is an exponent! when no base is indicated, the base is assumed to be 10 Common log x log10 x Logarithms log x y 10 y x log b n Change of Base log a n log b a Formula where a, b, and n are positive numbers, and a 1, b 1 instead of log, ln is used; these logarithms have a base of e Natural ln x log e x Logarithms ln x = y e y x all properties of logarithms also hold for natural logarithms Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Definition Examples logb 1 0 written exponentially: b0 = 1 logb b 1 written exponentially: b1 = b logb b x x written exponentially: bx = bx blog b x x , where x > 0 10 log 10 7 7 log 3 9 x log 3 9 log 3 x log b MN log b M log b N log 1 yz log 1 y log 1 z 5 5 5 2 log4 log4 2 log4 5 M 5 logb logb M logb N N 7 log8 log8 7 log8 x x log 2 6 x x log 2 6 log b M p log b M p log 5 y 4 4 log 5 y log 6 (3 x 4) log 6 (5 x 2) log b M logb N if and only if M=N (3x 4) (5x 2) 107 Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Definition Examples logb 1 0 written exponentially: b0 = 1 logb b 1 written exponentially: b1 = b logb b x x written exponentially: bx = bx blog b x x , where x > 0 10 log 10 7 7 log 3 9 x log 3 9 log 3 x log b MN log b M log b N log 1 yz log 1 y log 1 z 5 5 5 2 log4 log4 2 log4 5 M 5 logb logb M logb N N 7 log8 log8 7 log8 x x log 2 6 x x log 2 6 log b M p log b M p log 5 y 4 4 log 5 y log 6 (3 x 4) log 6 (5 x 2) log b M logb N if and only if M=N (3x 4) (5x 2) Common Errors: M logb M logb N logb log b M N log b M log b N log b N log b M cannot be simplified log b N log b M log b N log b MN logb (M N) logb M logb N log b (M N) cannot be simplified p log b M log b Mp (log b M) p log b M p (log b M)p cannot be simplified 108 College Algebra with Trig Name:____________________________________ Review- Exponential and Logarithmic Functions part 1 Date:_____________________________________ ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK! Write each expression in terms of simpler logarithmic forms: 4 s5 1 m5n3 (1) log b x 5 y (2) log b (3) log b (4) logb u7 c8 p Given loga n, evaluate each logarithm to four decimal places: (5) log 3 42 (6) log1 5 (7) log 6 0.00098 2 Solve each equation and round answers to four decimal places where necessary: (8) log 2 x 3 (9) log 5 4 log 5 x log 5 36 (10) 1000 75e0.5 x (11) log 6 x 2 1 1 (12) log7 x (13) logx 4 49 2 (14) 10x 27.5 (15) log x log5 log 2 log(x 3) (16) log x log 2 1 (17) log 4 x 3 (18) log 9 (5 x ) 3 log 9 2 (19) log 20 log x 1 (20) 2 1.002 4 x (21) e25 x 1.25 (22) log(x 10) log(x 5) 2 1 (23) log 6 216 log 6 36 log 6 x 2 109 College Algebra with Trig Name:____________________________________ Review- Exponential and Logarithmic Functions part 2 Date:_____________________________________ SHOW ALL WORK: (1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22 million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he need for this investment? (2) The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially: c. Express the number of guppies as a function of the time t. d. Use your answer from part (a) to find how many guppies are present after 1 week? e. Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies? 110 SHOW ALL WORK: (3) The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by i log 0.00235x . What is the intensity, to the nearest tenth, at a depth of 40 feet? 12 (4) Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula i D 10 log , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per i0 square meter is a standardized sound level. Use this information and formula to find the intensity of the sound at the concert. 111 SHOW ALL WORK: (5) How many years, to the nearest year, will it take the world population to double if it grows continuously at an annual rate of 2%. (6) Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two banks? (7) Determine how much time, to the nearest year, is required for an investment to double in value if interest is earned at the rate of 5.75% compounded quarterly. 112 TRIGONOMETRY UNIT NOTES PART 1 113 College Algebra with Trig Name:____________________________________ Lesson- Intro to Trigonometry Date:_____________________________________ Objective: To review special right triangles and learn the basics about trig ratios. DO NOW: 1) Find the measure of the hypotenuse of a right triangle if the legs are 1 and 2. 2) Given equilateral triangle ABC, find the length of the altitude to AB if BC=4. 3) What is the length of the diagonal of square ABCD if AB=5? __________________________________________________________________________________________ The Three Basic Trig Functions ex: Find the length of the missing side of the triangle and the exact value of the three trigonometric functions of the angle theta ( ) in the figures below: 1) 2) 5 7 12 11 3) 4) 3 7 2 1 5) Find the values of the three trigonometric functions for angle in standard position if a point with the coordinates (-3, -5) lies on its terminal side. 114 College Algebra with Trig Name:____________________________________ Lesson: The building blocks of trig functions Date:_____________________________________ Building Blocks of the Unit Circle Graph set up: Quadrantal Angles: Coterminal Angles: Examples: Find an angle that is COTERMINAL with each. *Note: it is often helpful to draw a diagram when solving these types of problems. 1. 100º 2. 650º 3. 405º 4. 400º 115 Reference Angles: Examples: Find an angle that is the REFERENCE ANGLE of each. *Note: it is often helpful to draw a diagram when solving these types of problems. 5. 100º 6. 650º 7. 405º 8. 400º Function of a positive acute angle: Examples: Express the given function as a function of a positive acute angle and, if possible, find the exact function value. 1. tan 225º 2. cos 100º 3. cos 405º 4. sin 650 º 5. tan (-120º) 6. cos 400º 116 College Algebra with Trig Name:____________________________________ Lesson: Finding one trig function exactly given another Date:_____________________________________ Objective: To be able to find one trig function exactly given another without solving for the angle. 3 7 DO NOW: If sin and cos determine the quadrant in which lies. 4 4 __________________________________________________________________________________________ Solving trig functions exactly Process: Examples: Given the value of sin or cos and the quadrant in which lies, find the value of the other function. 1 4 1. sin , Quadrant IV 2. cos , Quadrant II 2 5 24 5 3. cos , Quadrant I 4. sin , Quadrant II 25 12 117 College Algebra with Trig Name:____________________________________ Lesson- The Unit Circle Date:_____________________________________ Objective: To learn the meaning of a radian and to learn to create the unit circle. DO NOW: Find the reference angle that corresponds to an angle of 240º. __________________________________________________________________________________________ What is a unit circle? __________________________________________________________________________________________ Special Right Triangles 30-60-90 Right Triangles 45-45-90 Right Triangles __________________________________________________________________________________________ Use the area below to create the unit circle with degrees that are multiples of 30 and 45. Note the sine and cosine result for each. 118 Find the exact value of each expression without using a calculator: 1. tan 135 2. sin 150 3. cos 210 4. cos315 Practice: Express the given function as a function of a positive acute angle and, if possible, find the exact function value. 5. tan 225º 6. cos 100º 7. cos 405º 8. sin 650º 9. tan (-120º) 10. cos 400º Find the exact value of the given expression. 11. tan 135º + sin 330º 12. sin 300º + sin (-240º) 13. (sin 60º)(cos 150º) – tan (-45º) 119 Deg 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 Sin 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Cos 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Tan 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 Csc 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 Sec 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 Cot 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 120 College Algebra with Trig Name:____________________________________ Review- Basic Trig Test Date:_____________________________________ Objective: To review the following concepts in preparation for a test I. SOHCAHTOA II. Coterminal, reference, and quadrantals III. Finding exact trig values IV. Unit Circle MIXED PBLM SET Answer each of the following neatly and completely and show all work. NO CALCULATORS. 1. Find the exact value of each expression: a. cos 270º b. sin 90º 7 3. Without finding , find the exact value of tan if cos and sin 0 . 8 4. Find the values of the three trigonometric functions for angle in standard position if a point with the coordinates (-3, -5) lies on its terminal side. 5. Given the following triangle find the measure of angle exactly 3 6 121 8 6. Without finding , find the exact value of cos if sin and tan 0 . 9 7. Find the values of the three trigonometric functions for angle in standard position if a point with the coordinates (5, -4) lies on its terminal side. 8. Find the length of the missing side and the exact value of the three trigonometric functions of the angle in each figure: a) b) c) 3 5 8 2 12 7 7 d) e) 9 13 11 9. Find the exact value of each expression without using a calculator: a. sin 150 b. cos210 c. cos315 122 TRIGONOMETRY UNIT NOTES PART 2 123 College Algebra with Trig Name:____________________________________ Activity- What is a radian? Date:_____________________________________ Objective: To discover what a radian is. Follow the directions below and be sure to round each answer to the nearest ten-thousandth. Do all work on separate paper. 1. Use the paper provided to cut as many horizontal strips as you can. For convenience purposes, make sure each is about ¾ inch wide. 2. Use a piece of tape to tape the edges together to form a quasi-cylinder. 3. Trace the circular edge on another sheet of paper and estimate the center. 4. Measure the distance from the center to the circumference of the circle. 5. Unfurl the paper and measure its length. 6. Determine how many times the length of the radius goes into the distance found in step 5. 7. Make a table comparing your width of paper, distance from center to edge, and quotient. 8. Find the mean of all of your quotients. What does the mean represent? 9. How many radians are in the circumference of a circle? 10. 1 radian is approximately equal to ______________ degrees. 11. There are ________ radians in 180 degrees and there are ________ radians in 360 degrees. 12. In your own words, a radian is…? PRACTICE: As you do the practice problems, see if an equation for the conversion from radians to degrees and degrees to radians becomes apparent. Find the radian measure Find the degree measure for each degree measure: for each radian measure: 1. 720 1. 2 2. 90 2. 2 3. 45 3. 6 2 4. 60 4. 3 124 Deg 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 Rad 0 1 1 2 3 4 3 5 6 7 5 8 9 10 7 11 12 , , , , 6 6 4 6 6 6 4 6 6 6 4 6 6 6 4 6 6 2 4 6 8 4 4 4 4 Sin 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Cos 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Tan 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 Csc 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 Sec 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 Cot 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 125 College Algebra with Trig Name:____________________________________ Lesson- Reciprocal Trig Functions Date:_____________________________________ Objective: To learn about the reciprocal trig functions csc, sec, cot. DO NOW: Construct the unit circle in the space provided. __________________________________________________________________________________________ Reciprocal Trig Functions Examples: Find the exact values of the following trig functions. 1. sec 300 2. cot 270 3. csc (-210) 4. sec π 5. (sec 2π/3)(sin 2π/3) 6. cot (π/4) +csc (3π/4) 126 College Algebra with Trig Name:____________________________________ Lesson: Using radians to solve trig functions Date:_____________________________________ Objective: Use radians to solve trig functions DO NOW: Determine the exact value of sin (-45) without using the calculator. Sketch a figure and find the coordinates for each circular point: 8 5 1. 2. 3 6 7 11 3. 4. 6 3 Find the sine, cosine, and tangent of each radian measure: 3 5. 6. 7. 2 2 4 127 College Algebra with Trig Name:____________________________________ Lesson: Radians and Arc Length Date:_____________________________________ Objective: Discuss the relationship among central angles, radii and arc lengths. DO NOW: Determine the exact value of tan (-45) without using the calculator. __________________________________________________________________________________________ What is a central angle? What is an arc length? Relationship among central angle, radius and arc length: Examples: 1. Find the measure of a positive central angle that intercepts an arc of 14 cm on a circle of radius 5 cm. 2. Find the length of the arc intercepted by a central angle of 3.5 radians on a circle of radius 6 m. 3. A wheel of radius 18 cm is rotating at a rate of 90 revolutions per minute. a. How many radians per minute is this? b. How many radians per second is this? c. How far does a point on the rim of the wheel travel in one second? d. Find the speed of a point on the rim of the wheel in centimeters per second. 128 College Algebra with Trig Name:____________________________________ Lesson- Inverse Trig Functions Date:_____________________________________ Objective: To learn to use inverse trig functions to solve for an angle or angles 5 DO NOW: Find the exact value of sec 4 __________________________________________________________________________________________ What is an inverse trig function? What is it used for? Examples: 1. Write in the form of an inverse function: cos 2 2. Write in the form of an inverse function: cos 45 2 3. Solve by finding the value of x to the nearest degree: Sin 1 (1) x 1 4. Solve by finding the value of x to the nearest degree: Arc cos x 2 Find each value (put angles in radian measure). Round any decimals to the nearest hundredth. 3 3 5. Arc tan 3 6. cos 2Sin 1 2 129 College Algebra with Trig Name:____________________________________ Lesson: Law of Cosines Date:_____________________________________ Objective: solve triangles by using the Law of Cosines Law of Cosines: 1. Suppose a triangle ABC has side a = 4, side b = 7, and angle C = 54º. What is the measure of side C? 2. Suppose a triangle XYZ has sides of x = 5, y = 6, and z = 7. What is the measure of the angle across from the side of measure 6? 3. Suppose a triangle ABC has side b = 2, side a = 5, and angle B = 27º. Find the measure of side c. 4. Suppose a triangle ABC has side b = 4, side a = 5, and angle B = 27º. Find the measure of side c. Exit Ticket: Complete on separate paper and hand in when finished. 1. In a triangle PQR we have p = 8 and r = 11. Angle Q is 47º. What is the length of side q? 2. A triangle XYZ has sides x = 1, y = 2, and z = 2.5. What is the measure of angle Y? 130 College Algebra with Trig Name:____________________________________ Lesson- Forces and the Law of Cosines Date:_____________________________________ Objective: To determine the resultant force vector when given two force vectors and an included angle. DO NOW: If mA 30 , AC=5, and AB=7, solve the triangle. Find all sides to the nearest tenth and angles to the nearest degree. __________________________________________________________________________________________ Force- push or pull upon an object resulting from the object's interaction with another object. Vector- a quantity of force having both magnitude and direction. Examples: 1. Two forces separated by 52 degrees acts on an object at rest. The magnitude of the two forces are 32 Newtons and 17 Newtons. Find the resultant force vector to the nearest Newton. 131 2. A game of “Three Way Tug-O-War” is being played by a group of students. Two of the students are trying to gang up on the other. They believe that it will be easier to win if they increase the angle they create with the third person. Is that true? Justify your answer by providing examples. 3. Two fisherman have hooked the same fish and they are trying to cooperatively reel it in. The angle the fisherman make with the fish is 87 degrees. If the first fisherman’s line has a maximum tensile strength 223 Newtons and the second fisherman’s line has a maximum tensile strength of 401 Newtons and the fishermans’ lines are at maximum strain, what is the resultant force applied to the fish? 4. What is the angle separating two component force vectors whose magnitude are 15N and 17N respectively if the resultant vector is 21N? Exit Ticket Make up your own real-life Force scenario. Solve it and turn in on a separate piece of paper. THE MORE CREATIVE THE BETTER! 132 College Algebra with Trig Name:____________________________________ Lesson: Law of Sines, area of a triangle Date:_____________________________________ Objective: solve triangles by using the Law of Sines find the area of a triangle Law of Sines: Area of Triangles: (1) Given DEF where D = 29, E = 112, and d = 22: (a) Solve DEF, rounding answers to the nearest tenth (b) Find the area of DEF to the nearest tenth 133 (2) Given ABC where A = 13, B = 6520, and a = 35: (a) Solve ABC such that: (i) C is in DMS form (ii) b is rounded to the nearest tenth (iii) c is rounded to the nearest tenth 1 (b) Find the area of ABC, to the nearest tenth, using the formula K bc sin A 2 (3) Given GHJ where g = 45.7, H = 111.1, and J = 27.3: (a) Solve GHJ, rounding answers to the nearest tenth (b) Find the area of GHJ (to the nearest tenth) 134 College Algebra with Trig Name:____________________________________ WKST- Law of Sines/Cosines WP Date:_____________________________________ 1. A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the nearest tenth of a foot. 2 45 25 ft 2. A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened? 40 m 25 m 122 3. Using the picture seen to the right, and rounding to the nearest tenth of a meter, find the height of the tree. 110 23 120 m 135 College Algebra with Trig Name:____________________________________ Lesson: Determining the number of Distinct triangles (ambiguous case) Date:_____________________________________ Objective: To determine the number of distinct triangles that can be formed given an angle and two consecutive sides DO NOW: The sides of a triangle measure 6, 7, and 9. What is the largest angle in the triangle? __________________________________________________________________________________________ Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct triangles for which you can solve. There is a shortcut method to finding the number of distinct triangles that exist: Assume: Given two sides and one opposite angle: If a is acute: If a is obtuse: a b sin a no solution a b sin a one solution a b no solution b a b sin a two solutions a b one solution a b one solution Examples: How Many distinct triangles can be formed from the given information? 1. a 2 , b 3, mA 45 2. a 9, b 12, and mA 35 136 College Algebra with Trig Name:____________________________________ Lesson: Law of Sines- The Ambiguous Case Date:_____________________________________ Objective: solve triangles by using the Law of Sines (ambiguous case) find the area of a triangle DO NOW: The sides of a triangle measure 6, 7, and 9. What is the measure of the smallest angle in the triangle? Law of Sines: __________________________________________________________________________________________ Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct triangles for which you can solve. Showing all work, find all solutions for each ABC. If no solutions exist, write none. Round all answers to the nearest tenth. 1. A = 42, a = 22, b = 12 2. b = 50, a = 33, A = 132 3. a = 125, A = 25, b = 150 4. a = 32, c = 20, A = 112 5. b = 15, c = 13, C = 50 6. a = 12, b = 15, A = 55 137 College Algebra with Trig Name:____________________________________ Review- Trig Test #2 Date:_____________________________________ Objective: To prepare for a test on the following topics I. Reciprocal Trig Functions a. Secant b. Cosecant c. Cotangent II. Radians and Arc Length III. Inverse trig functions IV. Law of Sines a. Including Ambiguous Case V. Law of Cosines a. Including Forces VI. Area of a triangle Practice Problem Set 1. Two adjacent apartment buildings in Geometry Garden Estates share a triangular courtyard. They plan to install a new gate to close the courtyard that forms an angle of 1048 with one building and an angle of 4820 with the second building, whose length is 527 feet. a. Find, to the nearest tenth, the area of the courtyard. b. Find, to the nearest tenth, the length of this new gate. 2. A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the nearest tenth of a foot. 2 45 25 ft 138 3. A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened? 40 m 25 m 122 4. Using the picture seen to the right, and rounding to the nearest tenth of a meter, find the height of the tree. 110 23 120 m 5. Solve ABC if c = 49, b = 40, and A = 53 (round each answer to the nearest tenth) 6. Solve ABC, to the nearest tenth, if A = 50, b =12, and c = 14 & find the area. 139 7. Two forces act upon a body at rest. The first force, 35N, is separated from the second force, 52N, by 63 degrees. Determine the resultant force to the nearest Newton. 8. The resultant force acting on a body at rest is 90 pounds. If one of the component forces is 70 pounds and the other is 110 pounds, find the angle separating the resultant force from the larger of the component forces to the nearest ten minutes. 9. Find each of the following exactly: 5 a. sec b. cot c. csc( ) 300 6 4 140 TRIGONOMETRY UNIT NOTES PART 3 141 College Algebra with Trig Name:____________________________________ Lesson- Graphing sine and cosine functions Date:_____________________________________ Objectives: To construct graphs of the sine and cosine functions Definitions: Amplitude: Frequency: Period: Graphing Trigonometric Functions: (1) Graph y sin x in the interval -2 x 2 y 1 x -1 Period: x-intercepts: Domain: y-intercepts: Range: Maximum point: Minimum point: 142 (2) Graph y cos x in the interval -2 x 2 y 1 x -1 Period: x-intercepts: Domain: y-intercepts: Range: Maximum point: Minimum point: (3) Graph y 2 sin 2 x in the interval - x 2 y x 1 3 (4) Graph y 3 cos x in the interval x 2 2 y x 143 College Algebra with Trig Name:______________________________ HW- Sine and cosine graphs Date:_______________________________ Objective: find the amplitude and period to graph sine and cosine functions (1) y 2 sin x 2 x 2 y x (2) y 4 cos x 3 x 3 3 y x (3) y 1 cos 2x x 2 2 y x 144 (4) y 2 sin x 2 x 2 4 y x (5) y 1.5 cos 4x x y x 2 (6) y 3 sin x 2 x 2 2 y x 145 3 (7) y sin x x 2 y 1 x -1 (8) y 2 cos 4x x y x 2 1 1 (9) y sin x 4 x 4 2 2 y x 146 College Algebra with Trig Name:____________________________________ Lesson: Phase shifts and translations Date:_____________________________________ Objectives: find the phase shift for sine and cosine functions graph translations of sine and cosine functions Phase Shift: Translation: (1) Graph y sin x in the interval 0 x 4 y x (2) Graph y cos 2x in the interval 0 x 2 2 y x 147 (3) Graph y 3 cos x in the interval -2 x 2 2 y x x (4) Graph y 2 sin in the interval -2 x 2 2 8 y x (5) Graph y cos 2x in the interval -2 x 2 y x 148 (6) Graph y 2 cos x 4 in the interval 0 x 4 y x (7) Graph y 2 sin 2x 2 in the interval -2 x 2 y x x (8) Graph y 4 cos 6 in the interval 0 x 4 2 y x 149 College Algebra with Trig Name:______________________________ HW- Phase shift and translations Date:_______________________________ GRAPH THE FOLLOWING TRIGONOMETRIC FUNCTIONS WITHIN THE GIVEN INTERVAL: (1) y cos ( x ) 1 x 3 y x (2) y sin x 1 2 x 3 2 2 y x (3) y 2 sin x 2 2 x 2 2 y x 150 cos 4x x 1 (4) y 2 y x (5) y 3 sin x 2 x 2 4 y x 151 College Algebra with Trig Name:____________________________________ WKST- More practice with phase shift and translations Date:_____________________________________ Objectives: find the phase shift and vertical translation for sine and cosine functions graph translations of sine and cosine functions (1) Graph y 2 cos x 1 in the interval 0 x 4 y x (2) Graph y 2 sin x 2 in the interval -2 x 2 y x 152 x (3) Graph y 4 cos 6 in the interval 0 x 4 2 y x x (4) Graph y 3 cos 1 in the interval - x 2 2 y x 153 College Algebra with Trig Name:_______________________________ Lesson- Writing equations of sine and cosine functions Date:________________________________ Objectives: write the equations of sine and cosine functions given the amplitude, period, phase shift, and vertical translation Writing Trigonometric Functions: Write an equation of the sine function with each given amplitude and period: (1) amplitude = 4, period = 2 (2) amplitude = 35.7, period = 4 (3) amplitude = 0.8, period = 10 Write an equation of the cosine function with each given amplitude and period: 5 (4) amplitude = , period = 8 7 (5) amplitude = 0.5, period = 0.3 (6) amplitude = 17.9, period = 16 154 Write a sine function with each given period, phase shift, and vertical translation: (7) period = 2, phase shift = 0, vertical translation = -6 (8) period = , phase shift = , vertical translation = 0 2 8 (9) period = , phase shift = , vertical translation = 3 4 Write a cosine function with each given period, phase shift, and vertical translation: (10) period = 3, phase shift = , vertical translation = -1 (11) period = 5, phase shift = -, vertical translation = -6 (12) period = , phase shift = , vertical translation = 10 3 2 State the amplitude, period, phase shift, and vertical translation for each function: x (13) y 7.5 cos A= P= PS = VT = 3 1 (14) y sin 6x 8 A= P= PS = VT = 4 3 (15) y sin 2x 9 A= P= PS = VT = 5 4 (16) y cos 3 x A= P= PS = VT = 2 155 College Algebra with Trig Name:______________________________ HW- Writing the equation of sine and cosine functions Date:_______________________________ Write an equation of the sine function with each given amplitude and period: (1) amplitude = 6.7, period = 6 (2) amplitude = 0.5, period = Write an equation of the cosine function with each given amplitude and period: 3 (3) amplitude = , period = 0.2 7 1 2 (4) amplitude = , period = 5 5 Write a sine function with each given period, phase shift, and vertical translation: (5) period = 2, phase shift = , vertical translation = 12 2 (6) period = 8, phase shift = -, vertical translation = -2 Write a cosine function with each given period, phase shift, and vertical translation: (7) period = , phase shift = , vertical translation = -1 4 (8) period = 4, phase shift = , vertical translation = 5 8 156 SHOW ALL WORK: State the amplitude, period, phase shift, and vertical translation for each function: (9) y 2 cos 0.5x 3 A= P= PS = VT = 2 3 (10) y cos x A= P= PS = VT = 3 7 (11) y 3 sin 2x A= P= PS = VT = 2 1 x (12) y sin 6 A= P= PS = VT = 3 3 6 (13) y 4 sin 4 x 4 A= P= PS = VT = 4 x (14) y 8 sin 1 A= P= PS = VT = 2 8 157 College Algebra with Trig Name:__________________________________________________ Lesson- Graphing Tangent Date:___________________________________________________ Objective: To learn the proper technique for graphing the tangent function. The tangent curve is unlike the sine and cosine curves. 1. It is not a smooth continuous curve 2. There is no maximum or minimum height (goes on to ) 3. There are values that are undefined What are the tangent values in the interval 0 2 ? Radians 0 2 3 5 7 5 4 3 5 7 11 2 6 4 3 2 3 4 6 6 4 3 2 3 4 6 Degrees Tan (fraction) Tan (decimal) Which values for tan are undefined in the interval 0 2 ? What happens graphically when there is an undefined value? Graph y tan x in the interval 0 2 on the graph on the reverse: 158 College Algebra with Trig Name:______________________________ Lesson- Graphing Reciprocal Trig Functions Date:_______________________________ Objective- To learn how to graph the reciprocal of sine, cosine, and tangent. 1. Graph the reciprocal of a sine function: 2. Graph the reciprocal of a sine function: 3. Graph the reciprocal of a sine function: 159 College Algebra with Trig Name:______________________________ Review for test- Trig Graphs Date:_______________________________ Objective: To review for a test on: I. Graphing sine (including phase shift and translation) II. Graphing cosine (including phase shift and translation) III. Graphing tangent from [2 ,2 ] IV. Writing the equations of sine and cosine V. Graphing Reciprocal Trig Functions x 1. Graph y 2 sin in the interval -2 x 2 2 4 y x 2. Graph y 3 cos x in the interval -2 x 2 2 y x 3. y 2 sin x 2 2 x 2 4 y x 160 4. y cos x 4 x y x 5. y 2 sin 4 x 2 x y x x 6. y 2 cos 1 x 2 2 y x 7. Graph, on separate paper, the reciprocal of y=sin x and y=cos x. 161 TRIGONOMETRY UNIT NOTES PART 4 162 College Algebra with Trig Name:_____________________________ HW- Rational Expressions Date:_______________________________ Simplify each of the following and put all answers in simplest factored form. 5x 3x x 2 x 30 2 x 2 11 x 12 1) 2 2) x 5x 6 x 4 2 2 x 2 11 x 6 4x 2 4x 3 x x2 3 a2 b2 5 x 1 x2 1 3) 4) a b 2a b a 3x 2x 2 x 1 x 1 163 Trigonometry Unit: Formulas & Identities Pythagorean and Quotient Identities sin2 A + cos2 A = 1 sin A tan2 A + 1 = sec2 A tan A cos A cot2 A + 1 = csc2 A cos A cot A sin A Functions of the Sum of Two Angles sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B – sin A sin B tan A tan B tan(A B) 1 tan A tan B Functions of the Difference of Two Angles sin (A – B) = sin A cos B – cos A sin B cos (A – B) = cos A cos B + sin A sin B tan A tan B tan(A B) 1 tan A tan B Functions of the Double Angle sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A cos 2A = 2 cos2 A – 1 cos 2A = 1 – 2 sin2 A 2 tan A tan 2A 1 tan2 A Functions of the Half Angle 1 1 cos A sin A 2 2 1 1 cos A cos A 2 2 1 1 cos A tan A 2 1 cos A 164 College Algebra with Trig Name:______________________________ Lesson- Proving Trig Identities Date:_______________________________ Objective: To prove the validity of pythagorean, reciprocal and quotient identities. 5x 3x DO NOW: 2 x 5x 6 x 4 2 ____________________________________________________________________________________ Pythagorean Identities Identity: an equation that is true for all values of the variable 2 a a2 Ex: a 2 b 2 (a b)( a b) or 2 b b Pythagorean Identities 1st : cos2 sin 2 1 This identity is from the equation of a circle centered at (0,0) and whose radius is 1. From the previous identity, others can be obtained. 2nd : 1 tan 2 sec2 3rd: cot2 1 csc2 Proving Trig Identities The idea is to show both sides of the equation can be written in the same form. Helpful hints: 1. Always start with the most complicated side sin 2. Look for algebraic identity that can be applied (ex: tan or cos2 sin 2 1 ) cos 3. Try writing the expression in terms of sine and/or cosine. 4. If you get stuck on one side, try the other! 5. Never cross over the equal sign. Work with each side independently. *To prove that an equation is an identity, you must show that it is true for all values of the variable for each side of the equation. Examples: 165 1 1 1 1. Prove: sin cos sin cos2 2 2 2 2. Prove: sin 4 cos4 sin 2 cos2 3. Prove: tan cot csc sec csc x cot x 4. Prove: cot x csc x tan x sin x 166 College Algebra with Trig Name:____________________________________ CW/HW: Trig Identity Mixed Problem Set Date:_____________________________________ Objective: To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum, difference, double angle and half angle rules to verify identities. Verify each identity: 1. sec4 x 2 sec2 x tan 2 x tan 4 x 1 2. (1 cos x)(cscx cot x) sin x 1 tan y sec y cos 2 x cot x cos 2 x tan x 1 3. 4. 1 cot y csc y cos 2 x cot x cos 2 x tan x 1 tan y cot y 5. cos 2 x(1 sec 2 x) sin 2 x 6. sec y csc y 1 cos 2 x 1 1 7. cot x 8. 2 sec 2 x sin 2 x 1 sin x 1 sin x 167 College Algebra with Trig Name:____________________________________ HW: Trig Identity Mixed Problem Set Date:_____________________________________ Objective: To use algebraic/proportion techniques in conjunction with Pythagorean, reciprocal, sum, difference, double angle and half angle rules to verify identities. HW Verify each of the following neatly on separate paper. Be sure to show all steps for full credit. WILL BE COLLECTED AND GRADED! [30 point quiz] 1. cos x tan x sin x 2. cot x cos x sin x csc x 1 sin x cos x sin 2 x 2 sin x 1 1 sin x 3. 2 sec x 4. cos x 1 sin x cos2 x 1 sin x tan x cot x 3 cos2 z 5 sin z 5 3 sin z 2 5. 1 2 cos2 x 6. tan x cot x cos2 z 1 sin z 168 College Algebra with Trig Name:_______________________________ CW/HW- Proving Trig Identities #1 Date:________________________________ Verify each of the following identities: 1. sin x cot x cos x 2. cos x sin x 2 1 2 sin x cos x 3. cos x(tan x sin x cot x) sin x cos 2 x 4. cot x cos x sin x cscx 5. 1 cos x 1 cos x tan 2 x 6. cscx cos x cot x sin x cos2 x 7. tan 2 x sin 2 x tan 2 x sin 2 x 8. sec4 x 2 sec2 x tan 2 x tan 4 x 1 sin 2 x 2 sin x 1 1 sin x cos x 9. 10. tan x sec x cos 2 x 1 sin x 1 sin x tan x 1 1 sin x cos x 11. 12. 2 sec x sin x 2 tan x cos x 2 cos x 1 sin x 169 College Algebra with Trig Name:_______________________________ Lesson: Sum, difference and ½ angle identities Date:________________________________ Objective: use the sum, difference, and half-angle identities to evaluate trigonometric expressions cos2 y DO NOW: Prove: 1 sin y 1 sin y Sum and Difference Identities Half Angle Identities Double Angle Identities 170 Showing all work, complete the following chart to find the exact value of each trigonometric expression using the specified trigonometric identity: use a sum or difference identity use a half-angle identity cos 105 sin 75 tan 165 171 College Algebra with Trig Name:____________________________________ CW/HW- Trig Identities #2- Sum and Difference Date:_____________________________________ Verify each of the following identities: 1. sin( x y) sin( x y) 2 sin x cos y 2. cos(x y) cos(x y) 2 cos x cos y tan 2 x tan 2 y cos(x y ) 3. tan(x y) tan(x y) 4. cot x tan y 1 tan 2 x tan 2 y sin x cos y sin( x y ) sin( x y ) 5. tan x tan y 6. cot x cot y cos x cos y sin x sin y cos(x y ) sin( x y ) 7. 1 tan x tan y 8. tan( x y ) cos x cos y cos(x y ) 172 College Algebra with Trig Name:_______________________________ CW/HW- Trig Identities #3- Double and ½ angle Date:________________________________ Verify each of the following identities: 4 tan x 1. sin 4x 2 sin 2x cos2x 2. tan 2 x 2 2 tan 2 x 1 3. (sin x cos x) 2 1 sin 2 x 4. csc2 x sec x csc x 2 5. 2 csc2 x csc2 x tan x 6. cot x tan x 2 csc2x 4 cos 2 x 2 7. cot x tan x 8. cot x tan x 2 cot 2x sin 2 x sec2 x 1 tan 2 x 9. sec 2 x 10. cos 2 x 2 sec2 x 1 tan 2 x 173 Trigonometric Identities & Equations Definitions & Identities The following trigonometric identities hold for all values of where each expression is defined: 1 1 1 Reciprocal sin cos tan csc sec cot Identities 1 1 1 csc sec cot sin cos tan The following trigonometric identities hold for all values of where each expression is defined: Quotient Identities sin cos tan cot cos sin The following trigonometric identities hold for all values of where each Opposite-Angle expression is defined: Identities sin () sin cos () cos tan ( ) tan The following trigonometric identities hold for all values of where each Pythagorean expression is defined: Identities sin2 cos2 1 tan2 1 sec2 1 cot2 csc 2 If A and B represent the measures of two angles, then the following identities hold for all values of A and B: sin (A B) sin A cos B cos A sin B sin (A B) sin A cos B cos A sin B Sum & Difference cos (A B) cos A cos B sin A sin B Identities cos (A B) cos A cos B sin A sin B tan A tan B tan A tanB tan(A B) tan(A B) 1 tan A tan B 1 tan A tanB If represents the measure of an angle, then the following identities hold for all values of : Double-Angle 2 tan cos 2 cos2 sin2 tan 2 Identities sin 2 2 sin cos cos 2 2 cos2 1 1 tan2 cos 2 1 2 sin2 If represents the measure of an angle, then the following identities hold for all values of : 1 cos 1 cos 1 cos Half-Angle sin cos tan , cos 1 Identities 2 2 2 2 2 1 cos where the sign of the radical is determined by the quadrant in which lies 2 174 College Algebra with Trig Name:______________________________ Review- Trig Identity Test Date:_______________________________ Show All Work! Use the sum or difference identities to find the Use the half-angle identities to find the exact exact values of each trigonometric expression: values of each trigonometric expression: 1. cos74 cos44 sin 74 sin 44 5. cos112.5 tan 110 tan 50 6. sin 165 2. 1 tan 110 tan 50 7. tan105 3. cos345 4. sin 195 Verify Each Identity 1 1 cos x sin x 1. 2 sec2 x 2. 1 tan x 1 sin x 1 sin x cos x 1 cos 2 x 3. 2 csc2 x 1 4. ln tan x ln cot x sin 2 x 2 tan x sin 2 x 5. sin 2 x 6. cos4 x sin 4 x cos 2 x 2 tan x 1 sin x cos x 7. 2 sin x cos x 2 sin x cos x sin 2 x 3 3 8. cos x 1 sin x 1 tan 2 x 4 cos 2 x 2 cos 2 x cot x tan x 9. 1 tan 2 x 10. sin 2 x 175 1 cot x 2 1 sin 2 x csc x ln sec x tan x ln sec x tan x 11. 12. 1 sin x cos x 13. cos x 1 sin x 14. 2 sin 2 cos 2 cos 4 1 sin 4 cos( u v) tan 3t tan t 2 tan t tan u cot v 15. cos u sin v 16. 1 tan 3t tan t 1 tan 2 t cot x cot y tan( x y ) sin(m n) cot x cot y 1 17. tan m tan n 18. cos m cos n tan y sin y y 19. log(cos x sin x) log(cos x sin x) log cos 2 x 20. cos 2 2 tan y 2 1 sin x 21. (sec x tan x) 2 22. sec4 s tan 2 s tan 4 s sec2 s 1 sin x tan x cot x 1 23. 24. tan x cot x (sec x csc x)(sin x cos x) sec x csc x cos x sin x tan x sin x x cos 3 sin 3 2 sin 2 25. sin 2 26. 2 tan x 2 cos sin 2 27. 1 cos5x cos3x sin 5x sin 3x 2 sin 2 x 28. cos2 cot2 cot2 cos2 176 TRIGONOMETRY UNIT NOTES PART 5 177 College Algebra with Trig Name:____________________________________ Lesson- Solving Trig Equations Date:_____________________________________ Objective: To learn to solve trig equations in both degree and radian measure. DO NOW: Find the exact value of csc (2π/3) __________________________________________________________________________________________ Solving Trig Equations There are two types of problems that we could deal with. 1. Trig Equations of a linear form. 2. Trig Equations of a quadratic form. Process: Process: Example: 4 cos 3 4 Example: 2 sin 2 sin x 1 0 178 Practice Problems Solve each from 0 2 2 sin 2 3 1. 2 sin 5 4 sin 6 2. Tan2 Tan 3. 2 2 4. cos2 5 8 cos 5. 2 sin 2 5 sin 2 0 6. 3 sin 2 cos 1 7. 2 cos 1 sec 8. sin 2 2 cos tan 9. sin 2 1 0 179 180