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: Understanding the Question:
We are given that \(\sin \theta = \frac{\sqrt{3}}{2}\). We need to determine the value of the trigonometric expression \(2\sqrt{3} \cdot \cos \frac{\theta}{2}\).

Step 2: Key Formula or Approach:
1. Identify the standard angle \(\theta\) for which the sine value is \(\frac{\sqrt{3}}{2}\). For acute angles:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
2. Once \(\theta\) is found, divide it by 2 to get the angle \(\frac{\theta}{2}\).
3. Substitute this angle into \(\cos \frac{\theta}{2}\) and evaluate the final expression.

Step 3: Detailed Explanation:
1. We are given:
\[ \sin \theta = \frac{\sqrt{3}}{2} \]
Since we are dealing with standard trigonometric values, we know:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
By comparing both equations, we find the angle:
\[ \theta = 60^\circ \]
2. Now, find the half-angle \(\frac{\theta}{2}\):
\[ \frac{\theta}{2} = \frac{60^\circ}{2} = 30^\circ \]
3. Substitute \(\frac{\theta}{2} = 30^\circ\) into the target expression:
\[ E = 2\sqrt{3} \cdot \cos \left(\frac{\theta}{2}\right) \]
\[ E = 2\sqrt{3} \cdot \cos(30^\circ) \]
4. We know that the standard value of \(\cos 30^\circ\) is \(\frac{\sqrt{3}}{2}\). Substitute this in:
\[ E = 2\sqrt{3} \cdot \frac{\sqrt{3}}{2} \]
5. Simplify the expression:
The factor of 2 in the numerator and denominator cancels out:
\[ E = \sqrt{3} \cdot \sqrt{3} \]
\[ E = 3 \]
Thus, the value of the expression is 3.

Step 4: Final Answer:
The value of the expression is 3, which corresponds to option (A).