Question

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

Answer 1

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.