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MAT 106 – 040 TRIGONOMETRY LECTURE 33 WING HONG TONY WONG 9.3 — Further trigonometric identities Double-angle identities sin(2x) = 2 sin x cos x cos(2x) = cos2 x − sin2 x tan(2x) = 2 tan x 1−tan2 x Since sin2 x + cos2 x = 1, we also have cos(2x) = 2 cos2 x − 1 = 1 − 2 sin2 x. From these two identities, we can deduce the half-angle identities. Half-angle identities q x sin x2 = ± 1−cos 2 cos x2 = ± Sum-to-product identities sin x + sin y = 2 sin x+y cos 2 x+y cos x + cos y = 2 cos 2 cos q tan x2 = ± 1+cos x 2 x−y 2 x−y 2 q 1−cos x 1+cos x = 1−cos x sin x = sin x 1+cos x x−y sin x − sin y = 2 cos x+y sin 2 2 cos x − cos y = −2 sin x+y sin x−y 2 2 Product-to-sum identities sin x cos y = 21 [sin(x + y) + sin(x − y)] cos x cos y = 21 [cos(x + y) + cos(x − y)] cos x sin y = 21 [sin(x + y) − sin(x − y)] sin x sin y = − 12 [cos(x + y) − cos(x − y)] Example 1. Verify cot x sin 2x = 1 + cos 2x. Example 2. Write sin 3x in terms of sin x. Example 3. (9.3.4) If tan θ = 5 3 and sin θ < 0, find sin 2θ and cos 2θ. Example 4. (9.3.12) Write cos2 number. Example 5. (9.3.20) Write number. π 8 − 1 2 as a single trigonometric function or as a single 4 tan x cos2 x−2 tan x 1−tan2 x as a single trigonometric function or as a single Example 6. (9.3.34) Use a half-angle identity to find an exact value for cos x2 , given sin x = − 45 and 3π < x < 2π. 2 Date: Monday, November 11, 2013. 1 Example 7. (9.3.56) Verify cos x = 1−tan2 1+tan2 x 2 x 2 . Example 8. (9.3.74) Write sin 9B − sin 3B as a product of trigonometric functions. Example 9. Find the value of sin 67.5◦ − sin 22.5◦ . Example 10. Find the value of tan 22.5◦ . 2