Question

If \(\sin \theta = \frac{\sqrt{3}}{2}\), then the value of \(2\sqrt{3} \cdot \cos \frac{\theta}{2}\) is :

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

Hint:

Familiarizing yourself with the trigonometric tables for standard angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)) is highly recommended.
Recognizing immediately that \(\sin\theta = \frac{\sqrt{3}}{2}\) corresponds to \(\theta = 60^\circ\) allows you to calculate the half-angle and resolve the expression in under half a minute.

Answer 1

The Correct Option is A

Step 1: Understand what is being asked.
We are given $\sin\theta=\frac{\sqrt{3}}{2}$ and need $2\sqrt{3}\cdot\cos\frac{\theta}{2}$. Instead of first identifying $\theta=60^\circ$, let us find $\cos\theta$ and then use the half-angle formula for cosine directly.

Step 2: Find $\cos\theta$.
Using the identity $\sin^2\theta+\cos^2\theta=1$, and treating $\theta$ as an acute angle so $\cos\theta$ is positive:
\[ \cos\theta = \sqrt{1-\sin^2\theta} = \sqrt{1-\frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]
Step 3: Apply the half-angle formula for cosine.
The half-angle formula states:
\[ \cos\frac{\theta}{2} = \sqrt{\frac{1+\cos\theta}{2}} \]
Substitute $\cos\theta=\frac{1}{2}$:
\[ \cos\frac{\theta}{2} = \sqrt{\frac{1+\frac{1}{2}}{2}} = \sqrt{\frac{3/2}{2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
Step 4: Substitute into the required expression.
\[ 2\sqrt{3}\cdot\cos\frac{\theta}{2} = 2\sqrt{3}\cdot\frac{\sqrt{3}}{2} \]
The 2 in the numerator and denominator cancels:
\[ = \sqrt{3}\cdot\sqrt{3} = 3 \]
Final Answer:
The value of the expression is 3, matching option (A).
\[ \boxed{3} \]