Question

If \(\sqrt{3} \sin \theta = \cos \theta\), then value of \(\theta\) is

• A. \(\sqrt{3}\)
• B. \(60^\circ\)
• C. \(\dfrac{1}{\sqrt{3}}\)
• D. \(30^\circ\)

Hint:

Convert all terms to a single trigonometric ratio (like \(\tan \theta\)) to simplify.

Answer 1

The Correct Option is B

Given:
\[ \sqrt{3} \sin \theta = \cos \theta \]

Step 1: Divide both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\))
\[ \sqrt{3} \tan \theta = 1 \] \[ \tan \theta = \frac{1}{\sqrt{3}} \]

Step 2: Find \(\theta\) using known value of \(\tan \theta\)
\[ \tan 60^\circ = \sqrt{3}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \] So, \[ \theta = 30^\circ \]

Step 3: Verify the correct value using the original equation
Check if \(\theta = 30^\circ\) satisfies the equation:
\[ \sqrt{3} \sin 30^\circ = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \] \[ \cos 30^\circ = \frac{\sqrt{3}}{2} \] Both sides are equal.

Final Answer:
\[ \boxed{30^\circ} \]