What is the trigonometric identity for sin(A + B)?

The trigonometric identity for sin(A + B) is indeed expressed as sinAcosB + sinBcosA. This identity allows us to find the sine of the sum of two angles A and B by incorporating the sine and cosine functions of each of those angles.

To understand why this is the correct identity, consider that the sine function represents the y-coordinate in the unit circle. When you add angles A and B, you effectively rotate around the circle. According to the properties of sine and cosine, the components of these angles combine multiplicatively and additively, leading to the formulation of this identity.

The term sinAcosB corresponds to taking the sine of angle A while projecting it against the cosine of angle B, which accounts for its horizontal displacement. Conversely, sinBcosA takes the sine of angle B and projects it against the cosine of angle A. Adding these two contributions provides the complete vertical displacement, i.e., the sine of the combined angle (A + B).

This identity is useful in various applications, such as simplifying trigonometric expressions and solving equations involving sine functions. Understanding and applying this identity is fundamental in trigonometric computations in both pure and applied mathematics.