Question

How do you find the exact value of $\arctan (\sqrt{3})$ ?

Answer 1

In the given question, we have been given an inverse trigonometric function and knowledge of trigonometric function is very important to solve these types of questions as they are simply reverse of each other.

Complete step by step solution:
First, recognize that the domain of the function $\arctan (x)$ is $\dfrac{-\pi}{2} < x < \dfrac{\pi}{2}$.
Now as we know,
$\arctan (x)$ and $tan(x)$ are inverse functions.
Which simply means that $\arctan (tan(x))=x$ and $tan(\arctan (x))=x$.
We can see tangent and arctangent as "undoing" one another.
For example, we know that $\tan (\dfrac{\pi }{4})=1$ which means that $\arctan (1)=\dfrac{\pi }{4}$.
So, for the value of $\arctan (\sqrt{3})$ is essentially asking, the angle whose tan gives $\sqrt{3}$ .
Since $\tan (\dfrac{\pi }{3})=\sqrt{3}$ ,
We can reverse this with the arctan function to see that

$\arctan (\sqrt{3})=\dfrac{\pi }{3}$

Note:

1. In $tan(x)$, x is an angle
2. In $\arctan (x)$, x is the value of the tangent function
3. We can say that $tan(x)$ and $\arctan (x)$ are opposite to each other and this knowledge can be utilized in solving such questions.