Question

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

Answer 1

Step 1: Understanding the expression:
We are given the expression \( \sin A \cos B + \cos A \sin B \), and we need to evaluate it for \( A = 30^\circ \) and \( B = 45^\circ \).
This expression is a standard trigonometric identity for the sine of a sum of two angles, specifically:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] Thus, we can simplify the expression as:
\[ \sin A \cos B + \cos A \sin B = \sin(A + B) \]

Step 2: Substituting the values of \( A \) and \( B \):
Now, substitute \( A = 30^\circ \) and \( B = 45^\circ \) into the identity:
\[ \sin(30^\circ + 45^\circ) = \sin(75^\circ) \]

Step 3: Finding the value of \( \sin 75^\circ \):
Using a calculator or trigonometric table, we find:
\[ \sin 75^\circ \approx 0.9659 \]

Step 4: Conclusion:
Therefore, the value of \( \sin A \cos B + \cos A \sin B \) for \( A = 30^\circ \) and \( B = 45^\circ \) is approximately \( 0.9659 \).