Sum to Product Trigonometric Formulas

What are Sum to Product Formulas?

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

The Sum to Product Formulas

The primary sum to product formulas are:

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

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 sums or differences of trigonometric functions into products, which can often simplify calculations and expressions.

Example Calculation

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

\[ \begin{align*} \sin(30°) + \sin(60°) &= 2 \sin(\frac{30° + 60°}{2}) \cos(\frac{30° - 60°}{2}) \\[10pt] &= 2 \sin(45°) \cos(-15°) \\[10pt] &= 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} \\[10pt] &= \frac{\sqrt{6} + \sqrt{2}}{2} \end{align*} \]

Visual Representation

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