Question

How do you write $\tan a = b$ in the form of an inverse function?

Answer 1

Hint: When we are finding the inverse, we are actually finding the angle whose trigonometric ratio is known. In this question we have to find the inverse of a tangent function. So, actually we have to find the angle whose tangent ratio is given.

Complete step by step answer:
An inverse function is a function that reverses the effect of the original function. The inverse tangent is a function that reverses the effect of the tangent function.
 We know that with the tangent function, we can calculate the opposite side if we know the adjacent side and the angle of a right triangle.
 The inverse tangent formula is used to find the angle when the side opposite to that angle and adjacent side are known to us. The inverse of Tangent is represented by arctan or ${\tan ^{ - 1}}$.
In the above question, it is given that $\tan a = b$.
When we write an inverse function, we are identifying the angle, whose trigonometric ratio is known.
We shift tan to RHS,
$a = {\tan ^{ - 1}}b$
In this case as $\tan a = b$ , we have the angle a, whose tangent ratio is b.
An inverse function is written as ${\tan ^{ - 1}}b$, which is equal to a.
Or we can say ${\tan ^{ - 1}}b = a$.

Note:
 Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also termed as arcus functions, anti trigonometric functions or cyclometric functions.
We know that we can write the identity as-
${\tan ^{ - 1}} (\tan x)= x$