Question

How do you find the value for \[\arctan (0)\] or \[{{\tan }^{-1}}(0)\] ?

Answer 1

Hint: To calculate this let suppose the value of this inverse function be \[x\]. Then take the \[\tan \] both sides then using the property of inverse functions that is \[\tan ({{\tan }^{-1}}t)=t\] then just do simple algebraic operations.

Formula used: \[\tan ({{\tan }^{-1}}t)=t\]

Complete step by step solution:
First of all. Let suppose the value of \[{{\tan }^{-1}}(0)\] be \[x\]
\[\Rightarrow {{\tan }^{-1}}0=x\]
Now taking \[\tan \] function both sides
\[\Rightarrow \tan ({{\tan }^{-1}}0)=\tan x\]
And we know that \[\tan ({{\tan }^{-1}}t)=t\]
So, using this property
\[\Rightarrow 0=\tan x\]
\[\Rightarrow \tan x=0\]
Since \[x\] is the output of the function \[{{\tan }^{-1}}t\]
And also, the range of this function is \[[0,{}^{\pi }/{}_{2})\]
And we know that \[\tan 0=0\]
\[\Rightarrow \tan x=\tan 0\]
Now compare this value with the above-calculated value
\[\Rightarrow x=0\]

Hence the value of \[{{\tan }^{-1}}(0)=0\]

Note:
When we have to find the value of the inverse function just assume a variable to the output value and then use the appropriate properties of inverse trigonometric functions. You should remember the values of trigonometric functions with their respective domain.