# Find the value of $x$in the following:

$\tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ $

Last updated date: 25th Mar 2023

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Answer

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Hint: - Use $\sin 45^\circ = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$

Given equation is

$\tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ $

As we know

$\sin 45^\circ = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$, and $\sin 30^\circ = \dfrac{1}{2}$

Substitute these values in the given equation

$

\Rightarrow \tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ \\

\Rightarrow \tan x = \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{{\sqrt 2 }}} \right) + \dfrac{1}{2} \\

$

As we have ${\text{ }}\left( {\sqrt 2 \times \sqrt 2 = 2} \right)$

$

\Rightarrow \tan x = \left( {\dfrac{1}{2}} \right) + \left( {\dfrac{1}{2}} \right) \\

\Rightarrow \tan x = \dfrac{{1 + 1}}{2} = \dfrac{2}{2} = 1 \\

$

Now we know $\tan 45^\circ = 1$

$

\Rightarrow \tan x = 1 = \tan 45^\circ \\

\Rightarrow x = 45^\circ \\

$

So, this is the required answer.

Note: -In such types of questions the key point we have to remember is that always remember all standard angle values of ${\text{sin, cosine and tan}}$, then simplify the given equation using these values we will get the required answer.

Given equation is

$\tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ $

As we know

$\sin 45^\circ = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$, and $\sin 30^\circ = \dfrac{1}{2}$

Substitute these values in the given equation

$

\Rightarrow \tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ \\

\Rightarrow \tan x = \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{{\sqrt 2 }}} \right) + \dfrac{1}{2} \\

$

As we have ${\text{ }}\left( {\sqrt 2 \times \sqrt 2 = 2} \right)$

$

\Rightarrow \tan x = \left( {\dfrac{1}{2}} \right) + \left( {\dfrac{1}{2}} \right) \\

\Rightarrow \tan x = \dfrac{{1 + 1}}{2} = \dfrac{2}{2} = 1 \\

$

Now we know $\tan 45^\circ = 1$

$

\Rightarrow \tan x = 1 = \tan 45^\circ \\

\Rightarrow x = 45^\circ \\

$

So, this is the required answer.

Note: -In such types of questions the key point we have to remember is that always remember all standard angle values of ${\text{sin, cosine and tan}}$, then simplify the given equation using these values we will get the required answer.

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