5.5 Multiple-Angle and Product-to-Sum Formulas

5.5 Multiple-Angle and Product-to-Sum Formulas

Integrated Math 3 Honors

Mr. Washington

Multiple-Angle Formulas

• The first category involves functions of multiple angles such as sin ku and cos ku

• The second category involves squares of trigonometric functions such as $\sin^2u$  

• The third category involves functions of half-angles such as sin(u/2).

• The fourth category involves products of trigonometric functions such as sin u cos v.

Double-Angle Formulas

•  $\sin\ 2u=2\sin u\cos u$  

•  $\cos2u=\cos^2u-\sin^2u\ \ \ =2\cos^2u-1\ \ =1-2\sin^2u$  

•  $\tan2u=\frac{2\tan u}{1-\tan^2u}$  

Example 1

Use multiple-angle formulas to express cos 3x in terms of cos x

Power-Reducing Formulas

The double-angle formulas can be used to obtain the following power-reducing formulas.

Half- Angle Formulas

List the half-angle formulas

Example 2

Find the exact value of $\tan15\degree$  

Product-to-Sum Formulas

used to rewrite products of trig functions as a sum or difference

Example 3

Write cos 3x cos 2x as a sum or difference.

Sum-to-Product Formulas

Used to rewrite a sum or difference of trig function as a product

Example 4

Write cos 4x + cos 2x as a product