Question

What is $\arctan \left( \cos \pi \right)$?

Answer 1

Hint: For solving this question we can use the concept of trigonometric inverse functions. In this question we will take the value of $\cos \pi $ and then we will take the inverse of $\tan $ regarding that value. And then you will get some angle and that will be the solution for this question. And all questions of this type should be solved by this method.

Complete step by step solution:
According to our question we have to calculate the value of $\arctan \left( \cos \pi \right)$. For solving this question, we will use the trigonometric function concept and first we will solve the value which is given in the bracket and then we will solve the next step. So, as we can see that $\cos \pi $ is given in the bracket with ${{\tan }^{-1}}$ of $\arctan $, so we will solve that first. We know that the value of $\cos \pi =-1$. Now we will go to the next step and in that step, we have to take ${{\tan }^{-1}}$ of the value of $\cos \pi $. For taking the $\arctan $ we will always take a value which is in the range of $-\infty $ to $\infty $. It means that we can’t use the imaginary values in this.
Now, if we take $\arctan $of $\cos \pi $,
Then, $\arctan \left( \cos \pi \right)=\arctan \left( -1 \right)$
$\because \cos \pi =-1 \Rightarrow \arctan \left( \cos \pi \right)=-\dfrac{\pi }{4}$
Note that as a function the range of $\arctan $ is limited to $\left[ -\dfrac{\pi }{2},+\dfrac{\pi }{2} \right]$. Since $\arctan \left( -1 \right)$ means that the $\dfrac{\text{opposite}}{\text{adjacent}}$ for the angle must be (-1) or if we find it in form of $x$ and $y$ coordinates, then,
$\dfrac{y}{x}=-1\Rightarrow y=-x$
And we have an equilateral right angled triangle below the x-axis.
So, the value of $\arctan \left( \cos \pi \right)$ is equal to $-\dfrac{\pi }{4}$.

Note: During solving this question you should be careful about the values for which we are taking the $\arctan $ or ${{\tan }^{-1}}$ because it is compulsory that they must be a real value. They can’t be imaginary values. And they must be in the range of $\arctan \left( x \right)$, otherwise the solution of this question will go wrong.