Question

\( \sin 15^\circ = \)

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

Hint:

Memorize standard angle combinations like 15°.

Answer 1

The Correct Option is A

Concept: Use angle difference identity.

Step 1: Express: \[ 15^\circ = 45^\circ - 30^\circ \]

Step 2: Use identity: \[ \sin(A-B)=\sin A\cos B - \cos A\sin B \]

Step 3: Substitute values.
\[ \sin45=\frac{1}{\sqrt2}, \cos30=\frac{\sqrt3}{2} \] \[ \cos45=\frac{1}{\sqrt2}, \sin30=\frac{1}{2} \]

Step 4: Compute: \[ \frac{1}{\sqrt2}\cdot\frac{\sqrt3}{2} - \frac{1}{\sqrt2}\cdot\frac{1}{2} \]

Step 5: Simplify: \[ = \frac{\sqrt3-1}{2\sqrt2} \]