Multiple-Angle and Product-Sum Formulas Section 5.5 Multiple-Angle and Product-Sum Formulas

Objective By following instructions students will be able to: Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use power-reducing formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions. Use product-sum formulas to rewrite and evaluate trigonometric functions

Double-Angle Formulas Note:

Example 1: Solving a Multiple-Angle Equation Find all solutions of .

Example 2: Using Double-Angle Formulas in Sketching Graphs Analyze the graph of over the interval .

Example 3: Evaluating Functions Involving Double Angles Use the following to find , , and . , where

Example 4: Deriving a Triple-Angle Formula Express in terms of .

Power-Reducing Formulas

Example 5: Reducing a Power Rewrite as a sum of first powers of the cosines of multiple angles.

Half-Angle Formulas Note: The signs of sin(u/2) and cos(u/2) depend on the quadrant u/2 lies.

Example 6: Using a Half-Angle Formula Find the exact value of .

Example 7: Solving a Trigonometric Function Find all solutions of in the interval .

Product-to-Sum Formulas

Example 8: Writing Products as Sums Rewrite the product as a sum or difference.

Sum-to-Product Formulas

Example 9: Using a Sum-to-Product Formula Find the exact value of .

Example 10: Solving a Trigonometric Equation Find all solutions of in the interval .

Example 11: Verifying a Trigonometric Identity Verify the identity .

Revisit Objective Did we… Use multiple-angle formulas to rewrite and evaluate trigonometric functions? Use power-reducing formulas to rewrite and evaluate trigonometric functions? Use half-angle formulas to rewrite and evaluate trigonometric functions? Use product-sum formulas to rewrite and evaluate trigonometric functions?

Homework Pg 418 #1-79 EOO