Product to Sum Trigonometric Formulas

What are Product to Sum Formulas?

Product to sum formulas in trigonometry are identities that allow us to express the product of two trigonometric functions as a sum or difference of trigonometric functions. These formulas are essential in simplifying complex trigonometric expressions, solving equations, and performing integrations in calculus.

The Product to Sum Formulas

The primary product to sum formulas are:

\[ \begin{align*} \sin A \sin B &= \frac{1}{2}[\cos(A-B) - \cos(A+B)] \\ \cos A \cos B &= \frac{1}{2}[\cos(A-B) + \cos(A+B)] \\ \sin A \cos B &= \frac{1}{2}[\sin(A+B) + \sin(A-B)] \\ \cos A \sin B &= \frac{1}{2}[\sin(A+B) - \sin(A-B)] \end{align*} \]

Where:

• \(A\) and \(B\) are angles
• \(\sin\) and \(\cos\) are the sine and cosine functions, respectively

Derivation and Explanation

These formulas can be derived using the angle addition formulas and algebraic manipulation. They allow us to convert products of trigonometric functions into sums or differences, which can often simplify calculations and expressions.

Example Calculation

Let's calculate sin(30°)sin(60°) using the product to sum formula:

\[ \begin{align*} \sin(30°)\sin(60°) &= \frac{1}{2}[\cos(30° - 60°) - \cos(30° + 60°)] \\[10pt] &= \frac{1}{2}[\cos(-30°) - \cos(90°)] \\[10pt] &= \frac{1}{2}[\frac{\sqrt{3}}{2} - 0] \\[10pt] &= \frac{\sqrt{3}}{4} \end{align*} \]

Visual Representation

This diagram illustrates the angles 30° and 60° on the unit circle, visually representing the concept behind the product to sum formula for sin(30°)sin(60°).