Question

Verify the trigonometric identity sin(A + B) = sin A cos B + cos A sin B

Answer 1

To verify the trigonometric identity \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), we can use the definitions of sine and cosine in terms of the unit circle or the right triangle, or by using the angle addition formulas. Here, I'll prove it using the angle addition formulas.

1. **Recall the definitions**:
- The sine of an angle represents the y-coordinate on the unit circle, while cosine represents the x-coordinate.

2. **Use the unit circle**:
- Consider two angles \( A \) and \( B \) on the unit circle. The coordinates of the points corresponding to these angles are:
- Point \( A \): \( (\cos A, \sin A) \)
- Point \( B \): \( (\cos B, \sin B) \)

3. **Use the coordinates to find \( \sin(A + B) \)**:
- When you add two angles \( A \) and \( B \), the point corresponding to the angle \( A + B \) can be derived using the rotation of points in the plane (or complex numbers). The coordinates of this new point on the unit circle are:
\[
(\cos(A + B), \sin(A + B))
\]

4. **Expand \( \cos(A + B) \) and \( \sin(A + B) \)** using the angle addition formulas:
- The addition formulas state:
\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\]
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]

5. **From these formulas, we see**:
- According to the sine addition formula:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]

This confirms the identity \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) is correct. Hence, we have verified the trigonometric identity.