Find the value of $ x $ in each of the following:$ \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} $

Given:

\( \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \)

To do:

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

Solution:  

We know that,

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

$\sin 45^{\circ}=\frac{1}{\sqrt2}$

$\cos 45^{\circ}=\frac{1}{\sqrt2}$

Therefore,

\( \tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \)

$\Rightarrow \tan x=\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}+\frac{1}{2}$

$\Rightarrow \tan x=\frac{1}{2}+\frac{1}{2}$

$\Rightarrow \tan x=1$

$\Rightarrow \tan x=\tan 45^{\circ}$          (Since $\tan 45^{\circ}=1$) 

Comparing on both sides, we get,

$x=45^{\circ}$

Hence, the value of $x$ is $45^{\circ}$.

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Updated on: 10-Oct-2022

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