Question

How do find the exact value of ${\tan ^{ - 1}}0$ ?

Answer 1

In the above question you have to find the exact value of ${\tan ^{ - 1}}0$. You know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ , use this formula to solve the above problem. $\dfrac{{\sin \theta }}{{\cos \theta }}$ should also be zero for finding ${\tan ^{ - 1}}0$. So let us see how we can solve this problem.

Complete step by step solution:
In the given problem we need to find the exact value of ${\tan ^{ - 1}}0$ . Angle whose tangent is equal to zero is ${\tan ^{ - 1}}0$.
We know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$, so if $\tan \theta$ is zero than $\dfrac{{\sin \theta }}{{\cos \theta }}$ must be equal to zero. A fraction can only be zero if its numerator is zero. Therefore, $\sin \theta$ must be zero.
We know that the range of ${\tan ^{ - 1}}$ is $- \dfrac{\pi }{2}$ to $\dfrac{\pi }{2}$.
Therefore, we can say that the value of ${\tan ^{ - 1}}$ lies within the range.

The answer $\theta = \pi$ is not allowed, and the answer to the problem ${\tan ^{ - 1}}0 = 0$.

Note:
In the above solution we find the value of ${\tan ^{ - 1}}0$ is 0. Also, we did not consider $\theta = \pi$ because it will give us infinity, as $\sin \pi = 1$ and $\cos \pi = 0$. On dividing $\dfrac{{\sin \pi }}{{\cos \pi }}$ we get infinity. That’s why we considered $\theta = 0$.