Sum and Difference Notes Solution Key

Advanced Math Sum & Difference Identities Notes Sine & Cosine sin 30° = 1/2 cos 30° = √3/2 sin 60° = √3/2 cos 60° = 1/2 sin 45° = √2/2 cos 45° = √2/2 "α" and "β" are two angle measures. Sum and Difference Identities: cos(α ± β) = cosαcosβ ∓ sinαsinβ sin(α ± β) = sinαcosβ ± cosαsinβ Find the exact value of the following using sum and difference identities: 1. cos 75° cos(30° + 45°) = cos 30°cos 45° - sin 30°sin 45° = (√3/2)(√2/2) - (1/2)(√2/2) = √6/4 - √2/4 = (√6 - √2)/4 2. sin 15° sin(45° - 30°) or sin(60° - 45°) sin 45°cos 30° - cos 45°sin 30° (√2/2)(√3/2) - (√2/2)(1/2) √6/4 - √2/4 = (√6 - √2)/4 3. cos 15° cos(60° - 45°) or cos(45° - 30°) cos 60°cos 45° + sin 60°sin 45° (1/2)(√2/2) + (√3/2)(√2/2) √2/4 + √6/4 = (√2 + √6)/4 4. sin 105° sin(45° + 60°) sin 45°cos 60° + cos 45°sin 60° (√2/2)(1/2) + (√2/2)(√3/2) √2/4 + √6/4 = (√2 + √6)/4 Using reference angles, find the following: sin 135° = √2/2 cos 135° = -√2/2 unit circle with an angle of 135 degrees in the second quadrant cos 315° = √2/2 sin 315° = -√2/2 unit circle with an angle of 315 degrees in the fourth quadrant Now use #5 and #6 above, along with sum and difference identities to find the following: sin 195° (Use 60° and 135°) sin(60° + 135°) = sin 60°cos 135° + cos 60°sin 135° = (√3/2)(-√2/2) + (1/2)(√2/2) = -√6/4 + √2/4 = (-√6 + √2)/4 cos 345° (Use 30° and 315°) cos(30° + 315°) = cos 30°cos 315° - sin 30°sin 315° = (√3/2)(√2/2) - (1/2)(-√2/2) = √6/4 + √2/4 = (√6 + √2)/4 Draw a unit circle and label each of the quadrantal angles in degrees. unit circle with angles 0°, 90°, 180°, and 270° labeled Using the unit circle above and the sum and difference formulas, find the following: sin(90° - θ) sin(90° - θ) = sin 90°cosθ - cos 90°sinθ = 1*cosθ - 0*sinθ = cosθ cos(90° - θ) cos(90° - θ) = cos 90°cosθ + sin 90°sinθ = 0*cosθ + 1*sinθ = sinθ