Question

Evaluate: \[ \sin 20^\circ (4 + \sec 20^\circ) = ? \]

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

Hint:

To simplify trigonometric expressions, use standard identities like \( \sec x = \frac{1}{\cos x} \) and \( \frac{\sin x}{\cos x} = \tan x \). Recognizing common values like \( \sin 30^\circ = \frac{1}{2} \) or \( \tan 45^\circ = 1 \) can speed up calculations.

Answer 1

The Correct Option is A

We need to evaluate: \[ \sin 20^\circ (4 + \sec 20^\circ) \] 

Step 1: Express \(\sec 20^\circ\) in terms of \(\cos 20^\circ\) 
Since \[ \sec 20^\circ = \frac{1}{\cos 20^\circ} \] we rewrite the given expression as: \[ \sin 20^\circ \left( 4 + \frac{1}{\cos 20^\circ} \right) \]

 Step 2: Expand the expression 
Distribute \( \sin 20^\circ \): \[ 4 \sin 20^\circ + \sin 20^\circ \cdot \frac{1}{\cos 20^\circ} \] Using the identity: \[ \frac{\sin x}{\cos x} = \tan x \] we obtain: \[ 4 \sin 20^\circ + \tan 20^\circ \] 

Step 3: Substitute values of \(\sin 20^\circ\) and \(\tan 20^\circ\) 
From trigonometric tables: \[ \sin 20^\circ \approx 0.342 \] \[ \tan 20^\circ \approx 0.364 \] Substituting these: \[ 4(0.342) + 0.364 = 1.368 + 0.364 = 1.732 \] Since: \[ 1.732 = \sqrt{3} \] we conclude: \[ \sin 20^\circ (4 + \sec 20^\circ) = \sqrt{3} \] Thus, the correct answer is \( \mathbf{\sqrt{3}} \).