Section 6-4 Product Sum and Sum Product Identities

Transcription

1 480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 6-4 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing the product sum and sum product identities, which are easily derived from the sum and difference identities developed in Section 6-. These identities are used in calculus to convert product forms to more convenient sum forms. They also are used in the study of sound waves in music to convert sum forms to more convenient product forms. Product Sum Identities First, add left side to left side and right side to right side, the sum and difference identities for sine: or sin (x y) sin x cos y cos x sin y sin (x y) sin x cos y cos x sin y sin (x y) sin (x y) sin x cos y sin x cos y [sin (x y) sin (x y)] Similarly, by adding or subtracting the appropriate sum and difference identities, we can obtain three other product sum identities. These are listed below for convenient reference. PRODUCT SUM IDENTITIES sin x cos y [sin (x y) sin (x y)] cos x sin y [sin (x y) sin (x y)] sin x sin y [cos (x y) cos (x y)] cos x cos y [cos (x y) cos (x y)] A Product as a Difference Write the product cos 3t sin t as a sum or difference. Solution cos x sin y [sin (x y) sin (x y)] Let x 3t and y t. cos 3t sin t [sin (3t t) sin (3t t)] sin 4t sin t

2 6-4 Product Sum and Sum Product Identities 48 Solution Write the product cos 5 cos as a sum or difference. Finding Exact Values Evaluate sin 05 sin 5 exactly using an appropriate product sum identity. sin x sin y [cos (x y) cos (x y)] sin 05 sin 5 [cos (05 5 ) cos (05 5 )] [cos 90 cos 0 ] [0 ( )] 4 or 0.5 Evaluate cos 65 sin 75 exactly using an appropriate product sum identity. Sum Product Identities The product sum identities can be transformed into equivalent forms called sum product identities. These identities are used to express sums and differences involving sines and cosines as products involving sines and cosines. We illustrate the transformation for one identity. The other three identities can be obtained by following similar procedures. We start with a product sum identity: sin cos [sin ( ) sin ( )] () We would like x y Solving this system, we have x y x y () Substituting equation () into equation () and simplifying, we obtain sin x sin y cos x y All four sum product identities are listed on the next page for convenient reference.

3 48 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS SUM PRODUCT IDENTITIES sin x sin y sin x sin y cos x y cos x cos y cos x y cos x cos y cos x y cos x y 3 Solution A Difference as a Product Write the difference sin 7 sin 3 as a product. sin x sin y cos x y sin 7 sin 3 cos 73 cos 5 sin sin Solution Write the sum cos 3t cos t as a product. Finding Exact Values Find the exact value of sin 05 sin 5 using an appropriate sum product identity. sin x sin y cos x y sin 05 sin 5 cos 05 5 cos 60 sin 45 sin Find the exact value of cos 65 cos 75 using an appropriate sum product identity.

4 6-4 Product Sum and Sum Product Identities 483 Explore/Discuss The following proof without words of two of the sum product identities is based on a similar proof by Sidney H. Kung, Jacksonville University, that was printed in the October 996 issue of Mathematics Magazine. Discuss how the relationships below the figure are verified from the figure. y (cos, sin ) (cos, sin ) (t, s) sin sin cos cos s cos t cos sin cos x Answers to Matched Problems. cos 7 cos 3. (3 )/4 3. cos t cos t 4. 6/ EXERCISE 6-4 A In Problems 4, write each product as a sum or difference involving sine and cosine.. sin 3m cos m. cos 7A cos 5A 3. sin u sin 3u 4. cos sin 3 In Problems 5 8, write each difference or sum as a product involving sines and cosines. 5. sin 3t sin t 6. cos 7 cos 5 7. cos 5w cos 9w 8. sin u sin 5u B Evaluate Problems 9 exactly using an appropriate identity. 9. sin 95 cos cos 75 sin 5. cos 5 cos 75. sin 05 sin 65 Evaluate Problems 3 6 exactly using an appropriate identity. 3. cos 85 cos sin 95 sin cos 5 cos sin 75 sin 65 Use sum and difference identities to verify the identities in Problems 7 and 8.

5 484 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 7. cos x cos y [cos (x y) cos (x y)] 8. sin x sin y [cos (x y) cos (x y)] 9. Explain how you can transform the product sum identity sin u sin v [cos (u v) cos (u v)] into the sum product identity 0. Explain how you can transform the product sum identity cos u cos v [cos (u v) cos (u v)] into the sum product identity by a suitable substitution. Verify each identity in Problems 8. sin t sin 4t cos t cos 3t.. cos t cos 4t cot t sin t sin 3t tan t cos x cos y by a suitable substitution. cos x cos y cos x y cos x y sin x sin y cos x cos y cot x y sin x sin y cos x cos y tan x y cos x cos y sin x sin y cot x y cos x cos y sin x sin y tan x y cos x cos y cos x cos y cot x y sin x sin y sin x sin y tan [ (x y)] tan [ (x y)] Verify each of the following identities for the values of x and y indicated in Problems 9 3. Evaluate each side to five significant digits. (A) cos x sin y [sin (x y) sin (x y)] (B) cos x cos y cos x y cos x y 9. x 7.63, y x 50.37, y x.55, y x , y cot x y In Problems 33 40, write each as a product if y is a sum or difference, or as a sum or difference if y is a product. Enter the original equation in a graphing utility as y, the converted form as y, and graph y and y in the same viewing window. Use TRACE to compare the two graphs. 33. y sin x sin x 34. y cos 3x cos x 35. y cos.7x cos 0.3x 36. y sin.x sin 0.5x 37. y sin 3x cos x 38. y cos 5x cos 3x 39. y sin.3x sin 0.7x 40. y cos.9x sin 0.5x C Verify each identity in Problems 4 and cos x cos y cos z 4 [cos (x y z) cos (y z x) cos (z x y) cos (x y z)] sin x sin y sin z 4 [sin (x y z) sin (y z x) sin (z x y) sin (x y z)] In Problems 43 46, (A) Graph y, y, and y 3 in a graphing utility for 0 x and y. (B) Convert y to a sum or difference and repeat part A. 43. y cos (8x) cos (x) y cos (x) y 3 cos (x) 44. y sin (4x) sin (x) y sin (x) y 3 sin (x) 45. y sin (0x) cos (x) y cos (x) y 3 cos (x) 46. y cos (6x) sin (x) y sin (x) y 3 sin (x) APPLICATIONS Problems 47 and 48 involve the phenomena of sound called beats. If two tones having the same loudness and close together in pitch (frequency) are sounded, one following the other, most people would have difficulty in differentiating the two tones. However, if the tones are sounded simultaneously, they will interact with each other, producing a low warbling sound called a beat. Musicians, when tuning an instrument with other instruments or a tuning fork, listen for these lower beat frequencies and try to eliminate them by adjusting their instruments. Problems 47 and 48 provide a visual illustration of the beat phenomena. 47. Music Beat Frequencies. Equations y 0.5 cos 8t and y 0.5 cos 44t are equations of sound waves

6 6-5 Trigonometric Equations 485 with frequencies 64 and 7 hertz, respectively. If both sounds are emitted simultaneously, a beat frequency results. (A) Show that 0.5 cos 8t 0.5 cos 44t sin 8t sin 36t (The product form is more useful to sound engineers.) (B) Graph each equation in a different viewing window for 0 t 0.5: y 0.5 cos 8t y 0.5 cos 44t y 0.5 cos 8t 0.5 cos 44t y sin 8t sin 36t 48. Music Beat Frequencies. y 0.5 cos 56t and y 0.5 cos 88t are equations of sound waves with frequencies 8 and 44 hertz, respectively. If both sounds are emitted simultaneously, a beat frequency results. (A) Show that 0.5 cos 56t 0.5 cos 88t 0.5 sin 6t sin 7t (The product form is more useful to sound engineers.) (B) Graph each equation in a different viewing window for 0 t 0.5: y 0.5 cos 56t y 0.5 cos 88t y 0.5 cos 56t 0.5 cos 88t y 0.5 sin 6t sin 7t Section 6-5 Trigonometric Equations Solving Trigonometric Equations Using an Algebraic Approach Solving Trigonometric Equations Using a Graphing Utility The first four sections of this chapter consider trigonometric equations called identities. These are equations that are true for all replacements of the variable(s) for which both sides are defined. We now consider another class of trigonometric equations, called conditional equations, which may be true for some replacements of the variable but false for others. For example, cos x sin x is a conditional equation, since it is true for some values, for example, x /4, and false for others, such as x 0. (Check both values.) This section considers two approaches for solving conditional trigonometric equations: an algebraic approach and a graphing utility approach. Solving trigonometric equations using an algebraic approach often requires the use of algebraic manipulation, identities, and ingenuity. In some cases algebraic methods lead to exact solutions, which are very useful in certain contexts. Graphing utility methods can be used to approximate solutions to a greater variety of trigonometric equations, but usually do not produce exact solutions. Each method has its strengths. Explore/Discuss We are interested in solutions to the equation cos x 0.5 The figure at the top of the next page shows a partial graph of the left and right sides of the equation.