Question

If \[ 2 \sin 2\theta = \sqrt{3}, \] then \(\theta =\)?

• A. \(15^\circ\)
• B. \(27^\circ\)
• C. \(30^\circ\)
• D. \(40^\circ\)

Hint:

Divide both sides by constants to isolate \(\sin 2\theta\), then use inverse sine and consider the principal domain.

Answer 1

The Correct Option is C

Step 1: Isolate the sine term.
\[ 2\sin 2\theta = \sqrt{3} \implies \sin 2\theta = \frac{\sqrt{3}}{2} \]

Step 2: Recall where sine equals \(\frac{\sqrt{3}}{2}\).
The simplest angle in the first quadrant with this sine value is \(60^\circ\), so we take the principal case
\[ 2\theta = 60^\circ \]

Step 3: Solve for \(\theta\).
Dividing by 2 gives
\[ \theta = 30^\circ \]

Step 4: Confirm the answer.
Checking: \(2\sin(60^\circ) = 2\cdot\frac{\sqrt{3}}{2} = \sqrt{3}\), which matches the given equation.
\[ \boxed{30^\circ} \]