Sum, Difference, and Double Angle Formulas

In trigonometry, we often encounter angles that aren't on our standard unit circle, such as $15^\circ$ or $105^\circ$. The Sum and Difference Formulas allow us to evaluate these by expressing them as the sum or difference of angles we already know ($30^\circ, 45^\circ, 60^\circ$). A common mistake is thinking $\sin(A+B) = \sin A + \sin B$; however, trig functions are not distributive. Instead, we use specific patterns:

Notice the sign flip in the cosine formula! These identities are essential in physics for calculating the resulting phase of two overlapping waves.

What happens when $A = B$? The sum formulas transform into Double-Angle Formulas. These are powerful tools for reducing the degree of an expression or changing the frequency of a trigonometric function.

For cosine, there are three useful variations derived from the Pythagorean identity ($1 = \sin^2\theta + \cos^2\theta$): 1. $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ 2. $\cos(2\theta) = 2\cos^2\theta - 1$ 3. $\cos(2\theta) = 1 - 2\sin^2\theta$

Choosing the right version of $\cos(2\theta)$ can often turn a difficult calculus integral or physics problem into a simple one-step calculation.

In higher-level mathematics, we use these formulas for Power Reduction. By rearranging the double-angle formulas for cosine, we can rewrite $\sin^2\theta$ or $\cos^2\theta$ without the exponent. This is a critical step in Integral Calculus. Furthermore, in Acoustics, these formulas explain 'beats'—the pulsing sound you hear when two slightly different guitar strings are plucked simultaneously. Mastering these identities is less about memorization and more about recognizing patterns in the structure of waves.