MathematicsFind the value of cos 2θ if 2 sin 2θ=3.

Find the value of $\cos 2 θ$ if $2 \sin 2 θ = \sqrt{3}$ .

A
$\frac{3}{2}$
B
$\frac{5}{2}$
C
$\frac{1}{2}$
D
$\frac{7}{2}$

Solution:

Given,

2 \sin 2 θ = \sqrt{3}

Use the value

\sin 60^{\circ} = \frac{\sqrt{3}}{2}

. Evaluating the given expression.

2 \sin 2 θ = \sqrt{3}

\Rightarrow \sin 2 θ = \frac{\sqrt{3}}{2}

\Rightarrow \sin 2 θ = \sin 60^{\circ}

(\sin 60^{\circ} = \frac{\sqrt{3}}{2})

\Rightarrow 2 θ = 60^{\circ}

\Rightarrow θ = 30^{\circ}

Substitute

θ = 30^{\circ}

into

\cos 2 θ

.

\cos 2 (30^{\circ}) = \cos 60^{\circ}

\Rightarrow \cos 2 (30^{\circ}) = \frac{1}{2}

Therefore, the value of

\cos 2 θ

is

\frac{1}{2}

.
Hence, option 3 is correct.