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# Arctan

In trigonometry, Arctan means the inverse of tangent. There are six inverse trigonometric functions and Arctan is one of them. Arctan is also represented as arctangent (or) tan-1 (or) Tan-1. Arctan is an important study in trigonometry chapter. In this article, we will see about some of arctan formulas and solve some problems.

 Related Calculators Arctan Calculator

## Arctan Function

The inverse trigonometric functions are the inverse functions of the trigonometric functions with fixed domains. Arctan is the opposite of the tangent function and is used to calculate the angles of a right triangle. Let a, b, c and $\theta$ be the adjacent side, opposite side, hypotenuse and the angle of the right triangle respectively.

=> $\tan \theta$ = $\frac{b}{a}$

## Arctan Series

The derivative of the arctan is given as

=> $\frac{d}{dx}$ arctan x = $\frac{1}{1 + x^2}$

The infinite series for $\frac{1}{1 + x^2}$ can be found by using long division.

=> $\frac{1}{1 + x^2}$ = 1 - $x^2 + x^4 - x^6$ + ............

By integrating this infinite series, we get

$\int$($\frac{1}{1 + x^2}$) = $\int$(1 - $x^2 + x^4 - x^6$ + ............)

=> arctan x = x - $\frac{x^3}{3}$ + $\frac{x^5}{5}$ - $\frac{x^7}{7}$ + ............

which is the required infinite series for arctan.

## Calculate Arctan

Given below are the some of the examples in calculating arctan.

### Solved Examples

Question 1: Find the angle of right triangle for the given opposite side length is 5 cm and adjacent side is 12 cm?
Solution:
Opposite side = 5 cm

$\tan \theta$ = $\frac{\text{Opposite side}}{\text{Adjacent side}}$

$\tan \theta$ = $\frac{5}{12}$

$\tan \theta = 0.4167$

Take inverse on both sides.

$\theta = \arctan (0.4167)$

$\theta$ = 22.6°

The angle is 22.6°.

Question 2: Find the angle of right triangle for the given opposite side length is 20 cm and adjacent side is 25 cm?
Solution:
Opposite side = 20 cm

$\tan \theta$ = $\frac{\text{Opposite side}}{\text{Adjacent side}}$

$\tan \theta$ = $\frac{20}{25}$

$\tan \theta = 0.9$

Take inverse on both sides.

$\theta = \arctan (0.8)$

$\theta$ = 38.7°

The angle is 38.7°.

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