9-3: Other Identities

9-3: Other Identities Double-Angle Identities Notes: sin 2x = 2 sin x cos x cos 2x = cos2 x – sin2 x tan 2x = Notes: You never really need the tan 2x identity, because tan 2x = sin 2x/cos 2x It’s also why we never bothered with the tan (x + y) identities yesterday

9-3: Other Identities Ex 1: Use Double-Angle Identities If and , find sin 2x and cos 2x Since , we’re in the 3rd quadrant. In the 3rd quadrant, sin is negative, cos is negative Draw a triangle. -8 θ 17

9-3: Other Identities Use Pythagorean Theorem to find the missing leg b = -15 (remember: 3rd quadrant) sin 2x = 2 sin x cos x 2 ● -15/17 ● -8/17 = 240/289 cos 2x = cos2 x – sin2 x (-8/17)2 – (-15/17)2 = 64/289 – 225/289 = -161/289 -8 θ -15 17

9-3: Other Identities Example 2: Use Double-Angle Identities Express f(x) = sin 3x in terms of powers of sin x and constants sin 3x = sin (x + 2x) = (sin x)(cos 2x ) + (sin 2x)(cos x) = (sin x)(cos2 x – sin2 x) + (2 sin x cos x)(cos x) = sin x cos2 x – sin3 x + 2 sin x cos2 x = 3 sin x cos2 x – sin3 x = 3 sin x (1 – sin2 x) – sin3 x = 3 sin x – 3 sin3 x – sin3 x = 3 sin x – 4 sin3 x

9-3: Other Identities Because cos 2x = cos2 x – sin2 x, we can use the Pythagorean Theorem to rewrite the identity of cos 2x to occasionally make solving/proving problems easier. There are three forms of cos 2x cos 2x = cos2 x – sin2 x cos 2x = 1 – 2 sin2 x cos 2x = 2 cos2 x – 1

9-3: Other Identities Assignment Page 600 – 601 Problems 23 – 30 (all) and problem 43

9-3: Other Identities Power-Reducing Identities sin2 x = cos2 x =

9-3: Other Identities Example 4: Express f(x) = sin4 x in terms of constants and first powers of cosine functions f(x) = sin4 x = sin2 x ● sin2 x = ● = =

9-3: Other Identities Half-Angle Identities The sign in front of the radical depends upon the quadrant in which x/2 lies.

9-3: Other Identities Example 5A: Find the exact value of cos (since , x = ) (since cos is negative, so is cos x/2)

9-3: Other Identities Example 5B: Find the exact value of sin (since , x = ) (since sin is positive, so is sin x/2)

9-3: Other Identities Alternate Half-Angle Identities for Tangent The alternate identities remove the need to determine the sign If and , find tan x/2

9-3: Other Identities Ex 6: Use Half-Angle Identity for Tangent Since , we’re in the 3rd quadrant. In the 3rd quadrant, sin is negative, cos is negative Draw a triangle. -2 θ -3

9-3: Other Identities Use Pythagorean Theorem to find the missing leg c = sin x = cos x = -2 θ -3

9-3: Other Identities Assignment Page 600 Show work Problems 1 – 11 (odd) Problems 31 – 35 (odd) Show work

9-3: Other Identities Product-to-Sum Identities sin x cos y = ½ [sin(x + y) + sin(x – y)] sin x sin y = ½ [cos(x – y) – cos(x + y)] cos x cos y = ½ [cos(x + y) + cos(x – y)] cos x sin y = ½ [sin(x + y) – sin(x – y)] Sum-to-Product Identities sin x + sin y = 2 sin cos sin x – sin y = 2 cos sin cos x + cos y = 2 cos cos cos x – cos y = -2 sin sin

9-3: Other Identities Ex 7: Use Sum-to-Product Identities Prove the identity: Numerator Formula: sin x + sin y = 2 sin cos sin t + sin 3t = 2 sin cos = 2 sin 2t cos (–t) Denominator Formula: cos x + cos y = 2 cos cos cos t + cos 3t = 2 cos cos = 2 cos 2t cos (–t) (Next slide)

9-3: Other Identities

9-3: Other Identities Page 600 Problems 13 – 22 (all)