5.5 Multiple Angle & Product-to-Sum Formulas Homework: Page 394, #3, 5, 7, 14, 32, 33, 40, 48, 57

Double Angle Formulas

Double Angle Formulas

Double Angle Formulas

Power Reducing Formulas

Half-Angle Formulas The signs of sine and cosine depend on which quadrant the angle ends up in.

Product-to-Sum Formulas

Sum-to-Product Formulas

Examples Example 1: Solve 𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠𝑥=0 2 𝑐𝑜𝑠 2 𝑥−1+𝑐𝑜𝑠𝑥=0 2 𝑐𝑜𝑠 2 𝑥+𝑐𝑜𝑠𝑥−1=0 2𝑐𝑜𝑠𝑥−1 𝑐𝑜𝑠𝑥+1 =0 2𝑐𝑜𝑠𝑥−1 =0 𝑎𝑛𝑑 𝑐𝑜𝑠𝑥+1 =0 𝑐𝑜𝑠𝑥= 1 2 𝑎𝑛𝑑 𝑐𝑜𝑠𝑥=−1 𝑥= 𝑐𝑜𝑠 −1 1 2 𝑎𝑛𝑑 𝑥= 𝑐𝑜𝑠 −1 −1 𝑥= 𝜋 3 +2𝑛𝜋, 𝑥= 5𝜋 3 +2𝑛𝜋 𝑥=𝜋+2𝑛𝜋

Example 2: Analyze the graph on the interval [0, 2π) 𝑦=3(1− 2𝑠𝑖𝑛 2 𝑥) 𝑦=3𝑐𝑜𝑠(2𝑢) Amplitude = 3, period = π Key points on the interval [0,π] are: 0, 3 , 𝜋 4 , 0 , 𝜋 2 , −3 , 3𝜋 4 , 0 , (𝜋,3)

Example 3: Find sin(2u), cos(2u), and tan(2u), given: 𝑠𝑖𝑛𝑢= 3 5 , 0<𝑢< 𝜋 2 𝑐𝑜𝑠𝑢= 4 5 𝑡𝑎𝑛𝑢= 3 4 tan 2𝑢 = 2∙ 3 4 1− 3 4 2 cos 2𝑢 = 4 5 2 − 3 5 2 sin 2𝑢 =2∙ 3 5 ∙ 4 5 = 24 25 = 16 25 − 9 15 = 3 2 1− 9 16 = 7 25 = 3 2 ∙ 16 7 = 24 7

Example 4: Derive a triple-angle formula for cos3x cos 3𝑥 =cos⁡(2𝑥+𝑥) =𝑐𝑜𝑠2𝑥𝑐𝑜𝑠𝑥−𝑠𝑖𝑛2𝑥𝑠𝑖𝑛𝑥 = 2 𝑐𝑜𝑠 2 𝑥−1 𝑐𝑜𝑠𝑥−(2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥)𝑠𝑖𝑛𝑥 =2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2 𝑠𝑖𝑛 2 𝑥𝑐𝑜𝑠𝑥 =2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2𝑐𝑜𝑠𝑥 (1−𝑐𝑜𝑠 2 𝑥) =2 𝑐𝑜𝑠 3 𝑥−𝑐𝑜𝑠𝑥−2𝑐𝑜𝑠𝑥+ 2 𝑐𝑜𝑠 3 𝑥 =4 𝑐𝑜𝑠 3 𝑥−3𝑐𝑜𝑠𝑥

Example 5: Rewrite tan4x as a quotient of first powers of the cosines of multiple angles. 𝑡𝑎𝑛 4 𝑥= 1−𝑐𝑜𝑠2𝑥 1+cos⁡2𝑥 1−𝑐𝑜𝑠2𝑥 1+cos⁡2𝑥 = 1−𝑐𝑜𝑠2𝑥−𝑐𝑜𝑠2𝑥+ 𝑐𝑜𝑠 2 2𝑥 1+𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠2𝑥+ 𝑐𝑜𝑠 2 2𝑥 = 1−2𝑐𝑜𝑠2𝑥+ 1+𝑐𝑜𝑠4𝑥 2 1+2𝑐𝑜𝑠2𝑥+ 1+𝑐𝑜𝑠4𝑥 2 = 2−4𝑐𝑜𝑠2𝑥+1+𝑐𝑜𝑠4𝑥 2 2+4𝑐𝑜𝑠2𝑥+1+𝑐𝑜𝑠4𝑥 2 = 3−4𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠4x 3+4𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠4𝑥

Example 6: Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of 15. NOTE: 15 is half of 30 and 15 lies in Quadrant I: 𝑠𝑖𝑛 30° 2 = 1−𝑐𝑜𝑠30 2 𝑐𝑜𝑠 30° 2 = 1+𝑐𝑜𝑠30 2 = 1− 3 2 2 = 1+ 3 2 2 = 2− 3 2 2 1 = 2+ 3 2 2 1 = 2− 3 2 ∙ 1 2 = 2+ 3 2 ∙ 1 2 = 2− 3 2 = 2+ 3 2

𝑡𝑎𝑛 30° 2 = 1−𝑐𝑜𝑠30° 𝑠𝑖𝑛30° = 1− 3 2 1 2 = 2− 3 2 1 2 = 2− 3 2  2 1 = 2− 3 2

Example 7: Rewrite the following product as a sum or difference: 𝑠𝑖𝑛5𝑥𝑐𝑜𝑠3𝑥 = 1 2 sin 5𝑥+3𝑥 +sin⁡(5𝑥−3𝑥) = 1 2 sin 8𝑥 +sin⁡(2𝑥)