Question

Evaluate the following expression: \[ \sin 21^\circ \cos 9^\circ - \cos 84^\circ \cos 6^\circ \]

• A. 1
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{3}{2} \)

Hint:

Use trigonometric identities to simplify products of sines and cosines before solving.

Answer 1

The Correct Option is B

We are given the expression: \[ \sin 21^\circ \cos 9^\circ - \cos 84^\circ \cos 6^\circ \] We can simplify this expression using trigonometric identities. Using the identity for the product of sines and cosines: \[ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \] Step 1: Simplify \( \sin 21^\circ \cos 9^\circ \): \[ \sin 21^\circ \cos 9^\circ = \frac{1}{2} [\sin(21^\circ + 9^\circ) + \sin(21^\circ - 9^\circ)] = \frac{1}{2} [\sin 30^\circ + \sin 12^\circ] \] Since \( \sin 30^\circ = \frac{1}{2} \), this becomes: \[ \frac{1}{2} \left[\frac{1}{2} + \sin 12^\circ \right] = \frac{1}{4} + \frac{1}{2} \sin 12^\circ \] Step 2: Simplify \( \cos 84^\circ \cos 6^\circ \): \[ \cos 84^\circ \cos 6^\circ = \frac{1}{2} [\cos(84^\circ - 6^\circ) + \cos(84^\circ + 6^\circ)] = \frac{1}{2} [\cos 78^\circ + \cos 90^\circ] \] Since \( \cos 90^\circ = 0 \), this simplifies to: \[ \frac{1}{2} \cos 78^\circ \] Step 3: Combine the terms: Now substitute the expressions back into the original equation: \[ \frac{1}{4} + \frac{1}{2} \sin 12^\circ - \frac{1}{2} \cos 78^\circ \] Using approximate values for the trigonometric functions: \[ \sin 12^\circ \approx 0.2079, \quad \cos 78^\circ \approx 0.2079 \] Thus, the expression becomes: \[ \frac{1}{4} + \frac{1}{2} \times 0.2079 - \frac{1}{2} \times 0.2079 = \frac{1}{4} \] Hence, the final answer is \( \frac{1}{4} \).