Question

Evaluate: \(\sin A \cos B + \cos A \sin B\); if \(A = 30^\circ\) and \(B = 45^\circ\).

Answer 1

Step 1: Understanding the trigonometric identity:

The expression \( \sin A \cos B + \cos A \sin B \) is the standard trigonometric identity for \( \sin(A + B) \). This identity is as follows:
\[ \sin A \cos B + \cos A \sin B = \sin(A + B) \]

Step 2: Substituting the values:

We are asked to find the value of \( \sin(30^\circ + 45^\circ) \). Using the identity, we can write this as:
\[ \sin(30^\circ + 45^\circ) = \sin 75^\circ \]

Step 3: Finding the value of \( \sin 75^\circ \):

Using a calculator or known trigonometric values, we find:
\[ \sin 75^\circ \approx 0.9659 \]

Step 4: Conclusion:

Thus, the value of \( \sin 75^\circ \) is approximately \( 0.9659 \).