Preparing for the SAT II Trigonometry Trigonometry Trigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms. 2 ©Carolyn C. Wheater, 2000 Trigonometry Topics  Radian Measure  The Unit Circle  Trigonometric Functions  Larger Angles ©Carolyn C. Wheater, 2000  Graphs of the Trig Functions  Trigonometric Identities  Solving Trig Equations 3 Radian Measure  To talk about trigonometric functions, it is ©Carolyn C. Wheater, 2000 helpful to move to a different system of angle measure, called radian measure.  A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle. 4 Radian Measure  There are 2 radians in a full rotation -- ©Carolyn C. Wheater, 2000 once around the circle  There are 360° in a full rotation  To convert from degrees to radians or radians to degrees, use the proportion degrees radians   360 2 5 Sample Problems  Find the degree  Find the radian measure equivalent 3 of radians. 4 measure equivalent of 210° degrees radians   360 2 210 r   360 2 360r  420 420 7 r  360 6 6 ©Carolyn C. Wheater, 2000 degrees radians   360 2 d 3 4   360 2 2d  270 d  135  The Unit Circle  Imagine a circle on the ©Carolyn C. Wheater, 2000 coordinate plane, with its center at the origin, and a radius of 1.  Choose a point on the circle somewhere in quadrant I. 7 The Unit Circle  Connect the origin to the ©Carolyn C. Wheater, 2000 point, and from that point drop a perpendicular to the x-axis.  This creates a right triangle with hypotenuse of 1. 8 The Unit Circle  The length of its legs are ©Carolyn C. Wheater, 2000 the x- and y-coordinates of the chosen point.  Applying the definitions of the trigonometric ratios to this triangle gives y x sin( )   y cos    x 1 1  is the angle of rotation 1 x y bg 9 The Unit Circle  The coordinates of the chosen point are the cosine and sine of the angle .  This provides a way to define functions sin() and cos() for all real numbers . ©Carolyn C. Wheater, 2000 y sin( )   y 1  The x cos    x 1 bg other trigonometric functions can be defined from these. 10 Trigonometric Functions sin( )  y 1 csc   y 1 sec   x bg bg  is the angle of rotation 1 x cos   x ©Carolyn C. Wheater, 2000 bg y y tan   x bg x cot   y 11 bg Around the Circle  As that point ©Carolyn C. Wheater, 2000 moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold. 12 Reference Angles  The angles whose terminal sides fall in ©Carolyn C. Wheater, 2000 quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I.  The acute angle which produces the same values is called the reference angle. 13 Reference Angles  The reference angle is the angle between the ©Carolyn C. Wheater, 2000 terminal side and the nearest arm of the xaxis.  The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis. 14 Quadrant II Original angle  For an angle, , in quadrant II, the reference angle is   In quadrant II, ©Carolyn C. Wheater, 2000 Reference angle is positive  cos() is negative  tan() is negative 15  sin() Quadrant III Original angle  For an angle, , in quadrant III, the reference angle is -  In quadrant III, ©Carolyn C. Wheater, 2000 Reference angle  sin() is negative  cos() is negative  tan() is positive 16 Quadrant IV  For an angle, , in Reference angle quadrant IV, the reference angle is 2  In quadrant IV, is negative  cos() is positive  sin() ©Carolyn C. Wheater, 2000 Original angle  tan() is negative 17 All Star Trig Class  Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants. Star ©Carolyn C. Wheater, 2000 All Sine is positive All functions are positive Trig Tan is positive Class Cos is positive 18 Graphs of the Trig Functions  Sine  The most fundamental sine wave, y=sin(x), has the graph shown.  It fluctuates from 0 to a high of 1, down to –1, and back to 0, in a space of 2. ©Carolyn C. Wheater, 2000 19 Graphs of the Trig Functions  The graph of y  a sin b x  h  k is cb g h determined by four numbers, a, b, h, and k. amplitude, a, tells the height of each peak and the depth of each trough.  The frequency, b, tells the number of full wave patterns that are completed in a space of 2. 2  The period of the function is b  The two remaining numbers, h and k, tell the translation of the wave from the origin. 20  The ©Carolyn C. Wheater, 2000 Sample Problem 5 4 3 2 1 2 1 1 2 ©Carolyn C. Wheater, 2000 3 4 5 1 2  Which of the following equations best describes the graph shown?  (A) y = 3sin(2x) - 1  (B) y = 2sin(4x)  (C) y = 2sin(2x) - 1  (D) y = 4sin(2x) - 1  (E) y = 3sin(4x) 21 Sample Problem 5 4 3 2 1 2 1 1 2 ©Carolyn C. Wheater, 2000 3 4 5 1 2  Find the baseline between the high and low points.  Graph is translated -1 vertically.  Find height of each peak.  Amplitude is 3  Count number of waves in y = 3sin(2x) - 1 2  Frequency is 2 22 Graphs of the Trig Functions  Cosine  The graph of y=cos(x) resembles the graph of y=sin(x) but is shifted, or translated,  units to 2 the left.  It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2. 23 ©Carolyn C. Wheater, 2000 Graphs of the Trig Functions  The values of a, b, h, and k change the shape and location of the wave as for the sine. y  a cos b x  h  k cb g h ©Carolyn C. Wheater, 2000 Amplitude Frequency Period Translation a b 2/b h, k Height of each peak Number of full wave patterns Space required to complete wave Horizontal and vertical shift 24 Sample Problem  Which of the following 8 6 4 2 2 1 1 2 equations best describes the graph? ©Carolyn C. Wheater, 2000 y = 3cos(5x) + 4  (B) y = 3cos(4x) + 5  (C) y = 4cos(3x) + 5  (D) y = 5cos(3x) +4  (E) y = 5sin(4x) +3  (A) 25 Sample Problem  Find the baseline  Vertical translation + 4  Find the height of 8 6 4 2 2 1 1 2 peak ©Carolyn C. Wheater, 2000  Amplitude = 5  Number of waves in 2  Frequency y = 5cos(3x) + 4 =3 26 Graphs of the Trig Functions  Tangent  The tangent function has a discontinuous graph, repeating in a period of . ©Carolyn C. Wheater, 2000  Cotangent  Like the tangent, cotangent is discontinuous. • Discontinuities of the cotangent are  2 units left of those for tangent. 27 Graphs of the Trig Functions  Secant and Cosecant  The secant and cosecant functions are the reciprocals of the cosine and sine functions respectively.  Imagine each graph is balancing on the peaks and troughs of its reciprocal function. ©Carolyn C. Wheater, 2000 28 Trigonometric Identities  An identity is an equation which is true for ©Carolyn C. Wheater, 2000 all values of the variable.  There are many trig identities that are useful in changing the appearance of an expression.  The most important ones should be committed to memory. 29 Trigonometric Identities  Reciprocal Identities 1 sin x  csc x ©Carolyn C. Wheater, 2000  Quotient Identities sin x tan x  cos x cos x cot x  sin x 1 cos x  sec x 1 tan x  cot x 30 Trigonometric Identities  Cofunction Identities  The function of an angle = the cofunction of its complement. sin x  cos(90  x)   ©Carolyn C. Wheater, 2000 sec x  csc(90  x) tan x  cot(90  x)  31 Trigonometric Identities  Pythagorean Identities  The fundamental Pythagorean identity ©Carolyn C. Wheater, 2000 sin x  cos x  1 2 2   Divide the first by sin2x Divide the first by cos2x 1  cot 2 x  csc2 x tan2 x  1  sec2 x 32 Solving Trig Equations  Solve trigonometric equations by following these steps:  If ©Carolyn C. Wheater, 2000 there is more than one trig function, use identities to simplify  Let a variable represent the remaining function  Solve the equation for this new variable  Reinsert the trig function  Determine the argument which will produce the desired value 33 Solving Trig Equations  To solving trig equations:  Use  Let ©Carolyn C. Wheater, 2000 identities to simplify variable = trig function for new variable the trig function the argument 34  Solve  Reinsert  Determine Sample Problem  Solve 3  3 sin x  2 cos2 x  0  Use the Pythagorean 3  3 sin x  2 cos2 x  0 identity • (cos2x = 1 - sin2x)  Distribute ©Carolyn C. Wheater, 2000 3  3 sin x  2 1  sin 2 x  0 3  3 sin x  2  2 sin 2 x  0 1  3 sin x  2 sin 2 x  0 2 sin 2 x  3 sin x  1  0 35 c h like terms  Order terms  Combine Sample Problem  Solve 3  3 sin x  2 cos2 x  0 2  Let t = sin x 2 sin x  3 sin x  1  0 2t 2  3t  1  0  Factor ©Carolyn C. Wheater, 2000 and solve. (2t  1)(t  1)  0 2t  1  0 t  1  0 2t  1 1 t 2 36 t 1 Sample Problem  Solve 3  3 sin x  2 cos2 x  0  Replace t = sin x. t t ©Carolyn C. Wheater, 2000 = sin(x) = ½ when = sin(x) = 1 when 6  x 2 x  or 5 6 5  , ,  So the solutions are x  6 6 2  37