Question

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

Answer 1

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.