Algebra 101 - Unit 16

Introduction

Product-to-Sum formulas allow you to rewrite the product of sine and cosine functions as a sum or difference of trigonometric functions. These formulas are particularly useful for simplifying integrals, solving equations, and analyzing wave functions.

Formulas

• sin A · sin B = ½ [cos(A − B) − cos(A + B)]
• cos A · cos B = ½ [cos(A − B) + cos(A + B)]
• sin A · cos B = ½ [sin(A + B) + sin(A − B)]
• cos A · sin B = ½ [sin(A + B) − sin(A − B)]

Strategy for Using Formulas

1. Identify the trigonometric product you need to simplify (sin·sin, cos·cos, sin·cos, cos·sin).
2. Choose the correct product-to-sum formula.
3. Substitute the angles and simplify to obtain a sum or difference expression.
4. Use the resulting expression for further calculations or integration.

Examples

Example 1 — sin·sin:

Simplify sin 50° · sin 20°: sin 50° · sin 20° = ½ [cos(50° − 20°) − cos(50° + 20°)] = ½ [cos 30° − cos 70°] ≈ ½ [0.866 − 0.342] = 0.262

Example 2 — cos·cos:

Simplify cos 40° · cos 10°: cos 40° · cos 10° = ½ [cos(40° − 10°) + cos(40° + 10°)] = ½ [cos 30° + cos 50°] ≈ ½ [0.866 + 0.642] = 0.754

Example 3 — sin·cos:

Simplify sin 30° · cos 20°: sin 30° · cos 20° = ½ [sin(30° + 20°) + sin(30° − 20°)] = ½ [sin 50° + sin 10°] ≈ ½ [0.766 + 0.174] = 0.470

Practice Problems

1. Simplify sin 70° · sin 40° using product-to-sum formulas.
2. Simplify cos 60° · cos 30° using product-to-sum formulas.
3. Simplify sin 45° · cos 15° using product-to-sum formulas.
4. Simplify cos 50° · sin 20° using product-to-sum formulas.
5. Rewrite sin 80° · sin 40° as a sum or difference of cosines.