Question

How do you calculate the \[\arcsin \left( {\dfrac{{\sqrt 2 }}{2}} \right)\] ?

Answer 1

Hint: The question is related to the inverse trigonometry topic. Here is this question to find the value of \[\arcsin \left( {\dfrac{{\sqrt 2 }}{2}} \right)\] . To find the exact value we use the table of trigonometry ratios for standard angles and hence find the solution for the question.

Complete step-by-step answer:
The sine, cosine, tangent, cosecant, secant and cotangent are the trigonometry ratios of trigonometry. It is abbreviated as sin, cos, tan, cosec, sec and cot. Here in this question, we have \[\arcsin \left( {\dfrac{{\sqrt 2 }}{2}} \right)\] , where arcsin represents the inverse of a sine function. So we have to find the \[\arcsin \left( {\dfrac{{\sqrt 2 }}{2}} \right)\] .
The 2 can be written as \[\sqrt 2 \times \sqrt 2 \] . So the above function is written as
 \[ \Rightarrow \arcsin \left( {\dfrac{{\sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }}} \right)\]
On simplifying
 \[ \Rightarrow \arcsin \left( {\dfrac{1}{{\sqrt 2 }}} \right)\]
To find the value we use the table of trigonometry ratios for standard angles.
The table of sine function for standard angles is given as

Angle	0	30	45	60	90
sin	0	\[\dfrac{1}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{{\sqrt 3 }}{2}\]	1

Now consider the given function
 \[\arcsin \left( {\dfrac{1}{{\sqrt 2 }}} \right) = x\]
This can be written as
 \[ \Rightarrow si{n^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right) = x\]
So taking the sine function we have
 \[ \Rightarrow \dfrac{1}{{\sqrt 2 }} = \sin x\]
From the table of sine function for standard angles and by the property of sine function we get
 \[ \Rightarrow x = {45^ \circ }\]
This is in the form of degree; let us convert into radians.
To convert the degree into radian we multiply the degree by \[\dfrac{\pi }{{180}}\]
Therefore, we have \[x = 45 \times \dfrac{\pi }{{180}}\]
On simplification we have
 \[ \Rightarrow x = \dfrac{\pi }{4}\]
Therefore, the exact value of \[\arcsin \left( {\dfrac{{\sqrt 2 }}{2}} \right)\] is \[\dfrac{\pi }{4}\] .
So, the correct answer is “ \[\dfrac{\pi }{4}\] ”.

Note: The trigonometry and inverse trigonometry are inverse for each other. The inverse of a function is represented as the arc of the function or the function is raised by the power -1. For the trigonometry and the inverse trigonometry we need to know about the table of trigonometry ratios for the standard angles.