Inverse Tan: Formula, Derivative and Graph

The inverse in Trigonometry for tangent is known as Inverse Tan. It is the basic function that we use in real-world problems and solve them. However, the primary use of Tan Inverse is to apply the tangent ratio for a specified angle. Thus, with the help of this function, you can quickly find any value to tangent. Examples are tan 1, tan 10, arctan 1 and more. Here arc is the basic way to name any inverse formula in trigonometry. Thus we can say that the inverse of the tangent is also known as arctan. In the article below, you will understand why and how to use this formula to solve different problems. Also, you will learn about its practical applications. 

Tan Inverse Formula

The inverse of tan or anti-tan is the arcus of tan. Here, we will define the formula of the tan inverse.

Suppose we are given as x =tan y

then, y =tan-1 x

Here y can be any real number. 

This is also known as tan inverse x. 

When you add two different tan inverse functions, the formula will be represented as below.

\[\tan a\pm b = \frac{tana\pm tanb}{1-\mp tan(a)tan(b)}\]

This is also known as the additional formula for inverse tan. It is derived from the addition of two inverses of tan. 

If in the above equation, we put a = arctan x and b = arctan y, we get the equation as presented below.

\[arc tan(x)\pm arctan(y) = arctan\left ( \frac{x\pm y}{1\mp xy} \right )(mod\pi ), xy\neq 1\]

This is achieved after substituting the values of a and b.

The Derivative of Tan Inverse

To find out the derivative for the inverse of tan, we will find the derivative for tan inverse x. 

The formula of derivative of the tan inverse is given by:

d/dx(arctan(x)). Hence, we define derivatives as 1/ (1 + x2). Here x does not belong to i or -i. This is also known as the differentiation of tan inverse.

Let us take an example for a graph of the tan inverse. 

We will define it with the help of the graph plot between π/2 and –π/2.

Graph of Tan Inverse x

(Image will be Uploaded Soon)

In the above graph of tan inverse x, the points are plotted between π/2 and –π/2. The plotting is along the real axis. 

Integration of Tan Inverse x

If we want to give the values of definite integral for the inverse of tan, we use the concept of integration. The derivative of tan inverse will be integrated back to get the normal value of tan inverse. However, for a definite point, the value will be fixed. Thus the below expression defines the integral of inverse tan.

\[arctan(x) = \int_{0}^{x}\frac{1}{y^{2}+1}dy\]

The above pictures describe the integral of tan inverse x.

The basic graph for tan inverse is given by

Image will be Uploaded Soon

The above graph is defined for tangent inverse give by equation 1/ (1 + x2)

How Can You Relate Tan Inverse Infinity With Other Trigonometric Equations?

We take an example of a triangle with one side as x and another as 1. The hypotenuse becomes 

√1+ x2

Suppose a triangle has an arctan angle as θ then we can define the below relation for finding other trigonometric functions:

Sin (arctan(x)) = x/ (√1+ x2)

Cos (arctan(x)) = 1 / (√1+ x2)

tan (arctan(x)) = x

What are The Different Properties of Inverse Tan?

Below are some mentioned properties of the tan inverse:

Suppose  = arctan (x)

Also, x = arctan (y)

Here we can use the real number as a domain.

Here -π/2 < y< π/2

This is the defined property of tan inverse used widely.

How Will You Give Value For tan-1 Infinity?

We know that a tan of 90 degrees is defined as infinity. Thus for tan-1 the value is 90 degrees. 

tan 90° = ∞ or tan π/2 = ∞

Hence,

\[\tan^{-1}(\infty ) = \frac{\pi }{2} or \tan^{-1}(\infty ) = 90^{\circ}\]