Sine of a Sum

This identity tells you how to find the sine of a combined angle without directly measuring the new angle. It comes from the geometry of rotations on the unit circle: when two angle changes are combined, their horizontal and vertical components mix in exactly the pattern sin A cos B + cos A sin B. In practice, it lets you break a complicated angle into simpler parts whose sine and cosine you already know.

Use this when an angle is written as a sum of two angles and you know, or can easily find, the sine and cosine of each part. It is especially useful in exact-value trig problems, simplifying expressions, and verifying identities.

• adding the terms as sin A sin B + cos A cos B instead of using sine–cosine pairs
• forgetting the minus sign when the angle is written as A - B
• using degree values as if they were radians, or vice versa