Chapter 5 Analytic Trigonometry 1
5.5 Multiple Angle Formulas Objective: Rewrite and evaluate trigonometric functions using: multiple-angle formulas power-reducing formulas half-angle formulas product-to-sum and sum-to-product formulas 2
Double Angles Use “sum” formulas to determine the double-angle formulas for sine, cosine, and tangent. sin 2u = sin (u + u) cos 2u = cos (u + u) tan 2u = tan (u + u) 3
Double-Angle Formulas 4
Note: Double-angle formulas are not restricted to the angles 2u and u. Other double combinations, such as 4u and 2u or 6u and 3u, are also valid. Two examples: sin 4u = 2 sin 2u cos 2u cos 6u = cos 2 3u – sin 2 3u 5
Example 1 Solve 2 cos x + sin 2x = 0. 6
Example 2 Use your graphing calculator to find the approximate solution(s) of the equation in the interval [0, 2π). If possible, find the exact solution(s) algebraically. 7
Example 3 Use the following to find sin 2θ, cos 2θ, and tan 2θ. 8
Example 4 Derive the triple-angle formula for sin 3x. Hint: sin 3x = sin (2x + x). 9
Power-Reducing Formulas 10
Example 5 Rewrite sin 4 x as a sum of first powers of the cosines of multiple angles. Hint: sin 4 x = (sin 2 x) 2 11
Alternate Forms Start with the power-reducing formula for sine. Replace u with u/2 to find the half-angle formula for sine. Similarly, we can derive the half-angle formulas for cosine and tangent. 12
Half-Angle Formulas How do we know whether to use + or – for sine and cosine? 13
Example 6 Find the exact value of sin 105 °. Hint: 105 ° is ½ of ______? 14
Example 7 Find all solutions in the interval [0, 2π). 15
Product-to-Sum Formulas – 1 16 We can verify these using sum and difference formulas – but we won’t.
Product-to-Sum Formulas – 2 17
Example 8 Rewrite the product as a sum or difference. cos 5x sin 4x 18
Sum-to-Product Formulas – 1 19
Product-to-Sum Formulas – 2 20
Example 9 Find the exact value of cos 195 ° + cos 105 °. 21
Example 10 Find all solutions in the interval [0, 2π). sin 5x + sin 3x = 0 22
Example 11 Verify the identity. 23
Homework 5.5 Worksheet