Sine of a Difference

This identity tells you how to find the sine of an angle formed by subtracting one angle from another without calculating the difference directly first. It comes from the geometry of rotating points on the unit circle, where the x- and y-components combine in a specific way when one angle is measured relative to another. The subtraction form is especially useful because it rewrites one trig expression into products of simpler sine and cosine terms.

Use this when expanding or simplifying trigonometric expressions that contain a difference inside a sine function. It is also useful in solving equations, proving identities, and evaluating exact values such as sin(75°) or sin(a-b) from known angle values.

• reversing the order and writing sin A cos B + cos A sin B
• mixing up the subtraction formula for sine with the one for cosine
• forgetting to keep the angle order consistent, which changes the sign of the result