Question

How do you solve $2(\sin {x^2}) = 1$ ?

Answer 1

Hint : Take out all the like terms to one side and all the alike terms to the other side. Take out all the common terms. Reduce the terms on the both sides until they cannot be reduced any further if possible. Then finally evaluate the value of the unknown variable.

Complete step by step solution:
First we will start off by taking all the like terms to one side.
$2({\sin ^2}x) = 1 \\\ {\sin ^2}x = \dfrac{1}{2} \;$
Now further we will take square root on both the sides.
${\sin ^2}x = \dfrac{1}{2} \\\ \,\sin x = \sqrt {\dfrac{1}{2}} \\\ \,\sin x = \dfrac{1}{{\sqrt 2 }} \;$
Then next we will take the inverse of the sine function.
$\,\sin x = \dfrac{1}{{\sqrt 2 }} \\\ x = {\sin ^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right) \;$
Now we know that the value of $\sin {45^0}$ is $\dfrac{1}{{\sqrt 2 }}$ . Hence, now we substitute the value of $\sin {45^0}$ and hence evaluate the value of $x$ .
$x = {\sin ^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right) \\\ x = \dfrac{\pi }{4} \;$
Hence, the value of $x$ is $\dfrac{\pi }{4}$ .
So, the correct answer is “ $\dfrac{\pi }{4}$ ”.

Note : The question belongs to the trigonometry topic. To find the value of x we must know the inverse trigonometry concept where trigonometric function and inverse trigonometric function will get cancelled. We must know about the table of trigonometric ratios for the standard angles.