What is the trigonometric identity for sin(α + β)?

The trigonometric identity for sin(α + β) is indeed given by the formula sin(α + β) = sin α cos β + cos α sin β. This identity is fundamental in trigonometry and is derived from the unit circle and the geometric interpretation of sine and cosine.

To understand why this is the correct expression, we can visualize angles α and β on the unit circle. When we add two angles, the coordinates of the resulting point can be expressed in terms of the sine and cosine of the original angles. Essentially, this identity decomposes the sine of the sum into the contributions of each angle.

The terms sin α cos β and cos α sin β represent how each angle's sine and cosine interact when combined. The sine of the sum of two angles relies on these products to maintain the properties of the sine function, such as periodicity and wave patterns. Therefore, option B captures the correct relationship while the other options do not reflect the behavior of sine addition accurately.