Inverse tangent function and it's formula

The inverse tangent function provides the inverse value of the tangent function. It can be denoted as arctan or tan-1. The arctan indicates the measure of radian of some arc located on the circle for particular value of the tangent. Suppose the tangent function tan x = y. In that case, the tan inverse function can be written as tan-1 y = x. Before going to the examples, we will have a quick summary on graph of inverse tangent.

Drawing the graph of inverse tangent

The inverse tan graph is given below which shows that it resembles inversely to that of the tangent graph.
graph of inverse tangent

It is evident from the graph that the range of inverse tangent is (-π/2, π/2). The domain of inverse tangent is (-∞, +∞). We know that the tangent of an angle is the ratio of the opposite to the adjacent side of a right triangle i.e. tanθ = (opposite / adjacent). If we take inverse on both the sides, the tangent angle θ is given by θ = tan-1(opposite / adjacent).

Quick look at inverse tangent formula

The inverse function in trigonometry that we have studied can be used for solving mathematical problems. Lets list out some of the important inverse tangent formula.

  • arctan x = 2 arctan(x/(1+ √(1+x2))).

  • tan(arctan x) = x.

  • tan x = y => x = arctan(y) + kπ.

This formulas can be used to solve the examples of inverse trig functions. Lets see couple of examples to have a better idea.

(1) Find inverse tangent of tan-1(1):

Recalling the values for tangent function, we already know that tan 45° = 1. So we need to take inverse on both the sides to get expected results. Tan-1(1) = 45°.

(2) What is tan(tan-1 0.8):

We know that tan(tan-1 x) = x. Comparing the question with the equation, we have the answer 0.8, i.e. tan(tan-1 0.8) = 0.8.

All the trig inverse functions finds wide variety of applications and inverse tangent is not lagging behind. So keeping it in option is not a choice for students.

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