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 \( \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ \) can be simplified using the sine addition formula, which states:

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

By setting \( A = 75^\circ \) and \( B = 15^\circ \), we can rewrite the original expression as:

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

Therefore,

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

Hence, the value of \( \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ \) is \( 1 \).

Answer 2

Step 1: Use the sine addition formula: \[ \sin A \cos B + \cos A \sin B = \sin (A + B) \] Step 2: Apply it to the given expression: \[ \sin 75^\circ \cos 15^\circ + \cos 75^\circ \sin 15^\circ = \sin (75^\circ + 15^\circ) = \sin 90^\circ \] Step 3: Evaluate \( \sin 90^\circ \): \[ \sin 90^\circ = 1 \] Hence, the value is \( {1} \).