Question

Find the value of $ \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ $.

• A. 1
• B. \(\frac{\sqrt{3}}{2}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{\sqrt{2}}{2}\)

Hint:

Use trigonometric addition formulas to simplify expressions involving sums of products of sines and cosines.

Answer 1

The Correct Option is A

The expression \( \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ \) simplifies via the sine addition formula:

\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]

Setting \( A = 75^\circ \) and \( B = 15^\circ \) yields:

\[\sin(75^\circ + 15^\circ)\]

This simplifies to:

\[\sin 90^\circ = 1\]

Consequently, \( \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ \) equals \( 1 \).