Question

Evaluate the following
${\tan ^{ - 1}}(\tan {12^o})$

Answer 1

By using the inverse trigonometry function of tangent, we can solve.
First, we are going to understand the given function and then identify what properties can be applied on the given function. Then, we apply the property ${\tan ^{ - 1}}(\tan \theta ) = \theta ,\,\,\,\,\,\dfrac{{ - \pi }}{2} < \theta < \dfrac{\pi }{2}$. Then we write the $\tan \theta$ in terms of $\tan (n\pi + \theta )$. By using the property, we get the required value.

Complete step by step answer:
We are given an inverse trigonometric function, which is
${\tan ^{ - 1}}(\tan 12)$
To solve the given inverse function, we need to apply properties of inverse trigonometric functions, such that we get the required value.
So, the property which we will use is ${\tan ^{ - 1}}(\tan \theta ) = \theta ,\,\,\,\,\,\dfrac{{ - \pi }}{2} < \theta < \dfrac{\pi }{2}$.
Before we apply this property, we are going to convert $\tan \theta$ in the form of $\tan (n\pi + \theta )$, such that we get a better evaluated value for the solution for the given function.
So,
We can write that ${12^o}$ in tangent function can be written as the same as $- 4\pi + 12$, which can be done by the help of understanding in which quadrant the tangent function lies in.
$\tan (12) = \tan ( - 4\pi + 12)$
Now, we will substitute this into the given function.
We get
${\tan ^{ - 1}}(\tan (12)) = {\tan ^{ - 1}}(\tan ( - 4\pi + 12))$
This is a better form for applying the property which we are going to use.
So, on applying we get apply
${\tan ^{ - 1}}(\tan ( - 4\pi + 12))$=$- 4\pi + 12$
Which implies
${\tan ^{ - 1}}(\tan 12)$ $= - 4\pi + 12$

Note: Instead of presenting the value of as${12^o}$, we have presented in the form of $- 4\pi + 12$, such that we get precise value in terms of $n\pi + \theta$ for better representation in the final solutions, we should have good knowledge of inverse trigonometry function properties for this problem.