Lesson 7.6 Exercises, pages

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1 Lesson 7.6 Exercises, pages A. Write each expression as a single trigonometric ratio. a) sin (u u) b) sin u sin u c) sin u sin u d) cos u cos u sin U cos U e) sin u sin u f) sin u sin u sin U 5. Determine the exact value of each expression. a) sin 5 cos 5 b) cos a sin a 6 b 6 b sin (5 ) sin 90 a b a b 6 c) cos tan 0 a 8 b d) - tan (0 ) cos a 8 b tan (0º) tan 0º cos tan 60º 6. Simplify each expression. a) cos u sin u b) sin u (cos U sin U) cos U c) cos u d) cos u (cos U ) sin U ( cos U ) cos U P DO NOT COPY. 7.6 Double-Angle Identities Solutions

2 B 7. Verify each identity for the given value of u. a) sin u sin u cos u; u 0 b) cos u cos u sin u; u 5 L.S. sin 60 R.S. cos U sin 0 cos 0 a a b b The left side is equal to the right side, so the identity is verified. L.S. cos U 0 cos cos 0 R.S. cos U sin U acos 5 b asin 5 b a b a b 0 The left side is equal to the right side, so the identity is verified. tan u 8. For the identity tan u, determine the non-permissible - tan u values of u over the set of real numbers, then verify the identity for u. cos U 0 cos U 0 tan U U k, U k, tan U k ç k ç U (k ), k ç U k, k ç To verify, substitute U L.S. tan U tan R.S. in each side of the identity. tan U tan U tan tan a b ( ) ( ) The left side is equal to the right side, so the identity is verified. 7.6 Double-Angle Identities Solutions DO NOT COPY. P

3 9. Given angle u is in standard position with its terminal arm in Quadrant and tan u - 5, determine the exact value of each trigonometric ratio. a) sin u b) cos u c) tan u Since the terminal arm of U lies in Quadrant, is positive. Use mental math and the Pythagorean Theorem to determine the value of r, which is 9. So, and cos U a) Substitute values of and cos U in: cos U a 9 ba 5 9 b b) Substitute the value of cos U in: cos U cos U a 5 b c) Substitute tan U in tan U tan U : 5 tan U tan U a 5 b a 5 b 50, or 9 9 5, or Prove each identity. a) cot u sin u sin u b) sin u (sin u cos u) L.S. cot U sin U a cos U sin U b cos U R.S. The left side is equal to the right side, so the identity is proved. R.S. ( cos U) sin U cos U cos U L.S. The left side is equal to the right side, so the identity is proved. sin u c) cot u d) tan u cos u sin u cos u - cos u R.S. cos U cos U sin U cos U cot U L.S. The left side is equal to the right side, so the identity is proved. L.S. tan U cos U a cos U cos U b cos U R.S. The left side is equal to the right side, so the identity is proved. P DO NOT COPY. 7.6 Double-Angle Identities Solutions

4 . Prove each identity. cos u - cos sin u a) u = -sin u sec u b) cot u cos u sin u cos u - cos U cos L.S. U L.S. cos U cos U cos U cos U cos U cos U sin U cos U cos U sin U sec R.S. U cos U cos U cot U cos U R.S. The left side is equal to the right side, so the identity is proved. cos U cos U The left side and the right side simplify to the same expression, so the identity is proved. cos sin u c) u d) + cos u = sec u - sin u = + cos u tan u + sin u tan u cos L.S. U cos U cos U cos U cos U cos U R.S. tan U cos U cos U ( cos U)(cos U) cos U ( cos U)(cos U) ()( cos U) cos U The left side and the right side simplify to the same expression, so the identity is proved. L.S. cos U cos U ( cos U) cos U cos U cos U cos U cos U cos U tan U R.S. sec U tan U tan U tan U tan U The left side and right side simplify to the same expression, so the identity is proved. 7.6 Double-Angle Identities Solutions DO NOT COPY. P

5 . Solve each equation over the domain 80 x < 80. a) cos x cos x b) sin x cos x cos x cos x 0 cos x cos x 0 ( cos x)(cos x ) 0 Either cos x 0 cos x 0 x 90 or x 90 Or cos x 0 cos x x 0 The roots are: x 0, x 90, and x 90 sin x sin x sin x sin x 0 ( sin x )(sin x ) 0 Either sin x 0 sin x x 0 or x 50 Or sin x 0 sin x x 90 The roots are: x 0, x 50, and x 90. Solve each equation over the domain < x <. a) sin x sin x 0 b) sin x cos x 0 ( ) sin x 0 sin x sin x 0 sin x ( cos x ) 0 sin x 0 Either sin x 0 sin x sin x 0 x or x or x or x 0 or x or x Or cos x 0 x cos x The roots are: x and 7 x x or x or x or 7 x The roots are: x 0, x, x, and x 7. For this solution of the equation sin x cos x over the domain 0 x <, identify the error then write a correct solution. sin x cos x cos x = cos x cos x cos x sin x sin x x or x sin x cos x cos x 0 cos x ( sin x ) 0 Either cos x 0 x or x Or sin x sin x 5 x or x 6 6 The roots are: x, x, 6 5 x, and x 6 In the rd line, cos x cannot be removed, because a possible solution is cos x 0. P DO NOT COPY. 7.6 Double-Angle Identities Solutions 5

6 5. Use algebra to solve the equation cos x cos x over the set of real numbers. Give the answer to the nearest hundredth. cos x cos x cos x cos x cos x cos x 0 b_ b ac Use: cos x Substitute: a, b, c _ a () ()() cos x () _ cos x _ cos x _ cos x >, so there is no solution for cos x. So, cos x Since is negative, the terminal arm of angle x lies in Quadrant or. The reference angle is: cos a b In Quadrant, x , or In Quadrant, x , or Verify by substituting each root in the given equation. The solution is: x.95 k, k ç or x. k, k ç Double-Angle Identities Solutions DO NOT COPY. P

7 6. A student said that the identity sin u sin u cos u could be extended so that sin u sin u cos u and sin 6u 6 sin u cos u. a) Is the student correct? If your answer is yes, explain why. If your answer is no, write correct identities for sin u and sin 6u. C cos U is not an identity. For example, suppose: U sin cos U sin cos a ba b sin 6U 6 cos U is not an identity. For example, suppose: U 6 sin 6U sin 6 cos U 6 sin cos sin 6a 0 ba b In cos U, replace U with U: cos U And, in cos U, replace U with U: sin 6U cos U b) Write an identity for sin bu,where b is a positive even number. b In cos U, replace U with bu, and replace U with U: sin bu cos b b U 7. Prove each identity. tan x + cot x cot x - tan x a) sec x tan x - cot x b) cot x tan x cot x cot x tan x R.S. R.S. tan x cot x cos x sin x cos x sin x cos x sin x cos x sin x sin x cos x sin x cos x cos x sin x cos x cos x sin x cot x L.S. cos x Since the left side is equal to the sec x right side, the identity is proved. L.S. Since the left side is equal to the right side, the identity is proved. P DO NOT COPY. 7.6 Double-Angle Identities Solutions 7

8 8. For each equation, determine the solution over the set of real numbers. a) b) cos 5x 0 ( ) sin (x) sin 6x sin 6x 6x k, k ç,or 6 5 6x k, k ç 6 So, x k, k ç,or 6 5 x k, k ç 6 ( cos 5x ) cos (5x) cos 0x 5 0x k, k ç,or 6 7 0x k, k ç 6 So, x k, k ç,or 5 7 x k, k ç Double-Angle Identities Solutions DO NOT COPY. P