Inverse Tan

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Tan Inverse

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

This is also known as the additional formula for inverse tan. It is derived for 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. 

arctan(x) \[\pm\] arctan(y) = arctan(\[\frac{x\pm y}{1 \mp xy}\]) (mod π), xy ≠ 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 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 (∞) = π/2 or tan-1 (∞) = 90°

FAQ (Frequently Asked Questions)

Q1. What are the Six Different Inverse Functions of Trigonometry?

In trigonometry, there are mainly six functions that are widely used. These include sine, cosine, cos, sectant, tangent, cosecant and cotangent. The use of these functions is to get the value of the sides of a triangle when the angle is given to us. However, in the case of the inverse of these six functions, the concept is reversed. The actual working of the inverse of all trigonometric functions is to find the angle of the triangle when sides are given to us. The concept of inverse trigonometric functions is widely operated in real-time examples, which include physics, navigation, engineering and geometry. The three primary notations for trigonometry that are widely in use are:

  1. tan-1 (x) 

  2. sin-1 (x)

  3. cosine-1 (x)

Q2. How Do You Define Tan x?

Usually, sin and cosine are the main trigonometric functions used in solving real-time problems. However, there are more functions that are generated by the ratio of both sin and cosine. There are four functions: cosecant, tangent, cotangent and secant. If you know the values of sin and cosine of an angle, then it is easy to find the values of other trigonometric equations. Therefore, the tangent of x is given by:


tan x = sin (x)/ cosine (x)


Similarly, you can find solutions to other functions as tan(x) = inverse of cotangent (x).

Secant (x) in inverse of cosine(x) and cosecant (x) is inverse of sin (x). With the given formulas, you can easily relate them to the concept of functions and thus find their domain, codomain, and range. Thus, tan x is valid when cosine (x) is not equal to zero. Hence the value of x must not be equal to π/2.