VIEWS: 19 PAGES: 20 POSTED ON: 9/16/2012
TRIGONOMETRIC IDENTITIES 1. Reciprocal, Quotient, and Pythagorean Identities – pg. 290-297 Assignment: pg. 296-297 #1-8, 10-14, 15 2. Sum, Difference, and Double-Angle Identities – pg. 299-308 Assignment: pg. 306-308 #1-11, 15-17, 20, 23 3. Proving Identities – pg. 309-315 Assignment: pg. 314-315 #1-8, 10-15, 16-18 4. Solving Trigonometric Equations Using Identities – pg. 316 – 321 Assignment: pg. 320-321 #1-11, 14-16, 17 5. Chapter Quiz 6. Chapter Review – pg. 322-323 Assignment: pg. 322-323 #1-21 7. Chapter Exam LESSON 1: RECIPROCAL, QUOTIENT, AND PHYTHAGOREAN IDENTITIES Learning Outcomes: Learn to verify a trigonometric identity numerically and graphically using technology Learn to explore reciprocal, quotient, and Pythagorean identities To determine non-permissible values of trigonometric identities To explain the difference between a trigonometric identity and a trigonometric equation What is the difference between an equation and an identity? is an equation. It is only true for certain values of the variable x. The solutions to this equation are -2 and 2 which can be verified by substituting these values into the equation. is an identity. It is true for all values of the variable x. Comparing Two Trigonometric Expressions 1. Graph the curves y = sin x and y = cos x tan x over the domain Graph the curves on separate grids using the same range and scale. What do you notice? 2. Use the table function on your calculator to compare the two graphs. Describe your findings. 3. Use your knowledge of tan x to simplify the expression cos x tan x 2 4. What are the non-permissible values of x in the equation sin x = cos x tan x? Given , the non-permissible values will occur when cos x = 0. That occurs at 90˚ and again at 270˚. 5. Are there any permissible values for x for which the expressions are not equal? The equations y = sin x and y = cos x tan x are examples of trigonometric identities. Trigonometric identities can be verified both numerically and graphically. Reciprocal Identities: 1 1 1 csc x sec x cot x sin x cos x tan x Quotient Identities: sin x cos x tan x cot x cos x sin x Verifying Identities for a Particular Case: When verifying an identity we must treat the left side (LS) and the right side (RS) separately and work until both sides represent the same value. This technique does not prove that an identity is true for all values of the variable – only for the value of the variable being verified. sin x Ex. Verify that tan x . cos x a. Determine the non-permissible values, in degrees, for the equation. 3 b. Verify that θ= 60˚. L.S. R.S. sin 60 tan 60 cos 60 3 3 (using exact value chart) 2 1 2 3 2 2 1 3 LS = RS c. Verify graphically that this is an identity. Ex. a. Determine the non-permissible values, in radians, of the variable in the expression The functions sec x and tan x both have non-permissible values in their domains: The value of tan x also cannot equal 0. Tan θ =0 at π Therefore, combine the restriction is . A restriction will occur every 90˚. 4 b. Simplify the expression. Pythagorean Identity Recall that point P on the terminal arm of an angle θ in standard position has coordinates (cos θ, sin θ). Consider a right triangle with a hypotenuse of 1 and legs of cos θ and sin θ. y P(cosθ, sinθ) x The hypotenuse is 1 because it is the radius of the unit circle. Apply the Pythagorean Theorem in the right triangle to establish the Pythagorean Identity: Pythagorean Identities: sin 2 x cos2 x 1 1 tan 2 x sec2 x 1 cot 2 x csc2 x These identities can be written in several ways: sin 2 x 1 cos2 x cos2 x 1 sin 2 x tan 2 x sec2 x 1 cot 2 x csc2 x 1 5 Ex. Verify the identity for the given values: a. L.S. = R.S. = b. 240˚(QIII) will have a ref angle of 60˚, therefore the Tan will be positive and sin will be negative. L.S. = R.S. = 6 b. Use quotient identities to verify the Pythagorean identity Assignment: pg. 296-297 #1-8, 10-14, 15 7 LESSON 2: SUM, DIFFERENCE, AND DOUBLE-ANGLE IDENTITIES Learning Outcomes: Learn to apply sum, difference, and double-angle identities to verify the equivalence of trigonometric expressions To verify a trigonometric identity numerically and graphically using technology Ex. Use exact values to verify the following statement: a. sin 60 30 sin 60 cos30 sin30 cos60 L.S. = R.S. = Sum and Difference Identities: sin( A B ) sin A cos B cos A sin B sin( A B ) sin A cos B cos A sin B cos( A B ) cos A cos B sin 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 tan A tan B tan( A B) 1 tan A tan B These are on the formula sheet Ex. Write the expression as a single trigonometric function. Has the same form as the right side of cos(A-B): 8 Ex. Find the exact value of sin15 break angle into exact values sin15 sin 45 30 sin 45 cos30 sin30 cos 45 Ex. Find the exact value of When faced with radians, it may be easier to switch to degrees (165˚) Rewrite as a addition of two special angles (135 + 30) From special triangles For 135, it has a reference angle of 45 in QII (tan is negative). Tan 45 = 1 Simplify by combining fractions: 9 Double Angle Identities: sin 2 A 2sin A cos A cos 2 A cos 2 A sin 2 A 2 tan A tan 2 A 1 tan 2 A These identities are on the formula sheet Ex. Express each in terms of a single trigonometric function: a. 2 sin 4x cos 4x 1 1 b. cos 2 A sin 2 A 2 2 10 5 5 c. sin x cos x 2 2 1 5 5 2sin x cos x 2 2 2 1 5 sin 2 x 2 2 1 sin 5 x 2 (the 0.5 is added to the outside of the brackets to balance off the 2 added to the inside, the 2 is added to match the identity sin2A=2 sinAcosA) Ex. Determine an identity for cos 2A that contains only the sine ratio. Ex. Determine an identity for cos 2A that contains only the cosine ratio. Ex. Simplify the expression to one of the three primary trigonometric ratios Assignment: pg. 306-308 #1-11, 15-17, 20, 23 11 LESSON 3: PROVING IDENTITIES Learning Outcomes: To prove trigonometric identities algebraically To understand the difference between verifying and proving an identity To show that verifying the two sides of a potential identity are equal for a given value is insufficient to prove the identity To prove that an identity is true for all permissible values, it is necessary to express both sides of the identity in equivalent forms. One or both sides of the identity must be algebraically manipulated into an equivalent form to match the other side. You cannot perform operations across the equal sign when proving a potential identity. Simplify the expressions on each side of the identity independently. Hints in Proving an Identity: 1. Begin with the more complex side 2. If possible, use know identities given on the formula sheet, eg. try to use the Pythagorean identities when squares of trigonometric functions are involved. 3. If necessary change all trigonometric ratios to sines and/or cosines, eg. sin x 1 replace tan x by , or sec x by . cos x cos x 4. Look for factoring as a step in trying to prove an identity. 5. If there is a sum or difference of fractions, write as a single fraction 6. Occasionally, you may need to multiply the numerator or denominator of a fraction by its conjugate. It is usually easier to make a complicated expression simpler than it is to make a simple expression more complicated. 12 1 Ex. Consider the statement cos x sin x tan x cos x a. Verify the statement is true for x 3 L.S. R.S. b. Prove the statement algebraically: L.S. R.S. c. State the non-permissible values for x. Work in degrees. The only non-permissible values for either side is cos x ≠ 0. 13 Ex. Consider the identity a. Describe how to use a graphing calculator to verify the identity. b. Prove the identity L.S. R.S. Ex. Prove that is an identity for all permissible values of x. 14 Ex. Prove that is an identity. For what values of x is this identity undefined? For non-permissible values, we need to find we the equations cos x – 3 = 0 or 2cos x – 1 = 0 are undefined. No solution. Quad. I and IV (positive) Ref angle is 60˚ QI: 60˚ QIV: 360˚-60˚ = 300˚ Undefined for x = 60˚+360˚n x = 300˚+360˚n Assignment: pg. 314-315 #1-8, 10-15, 16-18 15 LESSON 4: SOLVING TRIGONOMETRIC EQUATIONS USING IDENTITIES Learning Outcomes: To solve trigonometric equations algebraically using know identities To determine exact solutions for trigonometric equations where possible To determine the general solution for trigonometric equations To identify and correct errors in a solution for a trigonometric equation Investigation 1. Graph the function over the domain . Describe the graph. 2. From the graph, determine an expression for the zeros of the function over the domain of all real numbers. 3. Algebraically solve over the domain. Compare your answers to the question above. Ref angle is 60˚(answers in QI and QIV) 16 4. Which method, graphic or algebraic, do you prefer to solve the above equation? To solve some trigonometric equations, you need to make substitutions using the trigonometric identities that you have studied in this chapter. This often involves ensuring that the equation is expressed in terms of one trigonometric function. Ex. Solve the following equation where 0 x 2 . 2cos2 x 3sin x 0 2(1 sin 2 x) 3sin x 0 Pythagorean identity 2 2sin 2 x 3sin x 0 Expand brackets 0 2sin 2 x 3sin x 2 Need to factor 2sin x 1sin x 2 1 sin x sin x 2 2 17 Ex. Solve for x as an exact value where Ex. Solve the equation algebraically over the domain Check for any non-permissible values: occur when cos x = 0, at 90˚, 270˚ The non-permissible values are not part of our solution set, therefore our solutions are x = 30˚, 150˚, and 180˚ 18 b. Verify your answer algebraically. Ex. Algebraically solve . Give general solutions expressed in radians. Occurs at 0 Ref angle is , neg in QII andQIII General Solutions: 19 Ex. Algebraically solve Give general solutions expressed in radians. Occurs at 0 No solution. Only non-permissible value occurs when cosx = 0, (90˚, 270˚) The non-permissible values do not change our original solution of 0 +2πn Unless the domain is restricted, give general solutions. Check that the solutions for an equation do not include non-permissible values from the original equation. Assignment: pg. 320-321 #1-11, 14-16, 17 20