Find the value of $ x $ in each of the following:$ \cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ} $

Given:

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

To do:

We have to find the value of \( x \).

Solution:  

We know that,

$\cos 60^{\circ}=\frac{1}{2}$

$\cos 30^{\circ}=\frac{\sqrt3}{2}$

$\sin 60^{\circ}=\frac{\sqrt3}{2}$

$\sin 30^{\circ}=\frac{1}{2}$

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

$\Rightarrow \cos 2 x=\frac{1}{2}\times\frac{\sqrt3}{2}+\frac{\sqrt3}{2}\times \frac{1}{2}$

$\Rightarrow \cos 2 x=\frac{\sqrt3}{4}+\frac{\sqrt3}{4}$

$\Rightarrow \cos 2 x=\frac{2\sqrt3}{4}$

$\Rightarrow \cos 2 x=\frac{\sqrt3}{2}$

$\Rightarrow \cos 2 x=\cos 30^{\circ}$          (Since $\cos 30^{\circ}=\frac{\sqrt3}{2}$)

Comparing on both sides, we get,

$2x=30^{\circ}$

$x=\frac{30^{\circ}}{2}$

$x=15^{\circ}$

Hence, the value of $x$ is $15^{\circ}$. 

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Updated on: 10-Oct-2022

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