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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.

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Arctan Function

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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.

=> $\tan \theta$ = $\frac{b}{a}$
Then, $\theta = \arctan$$\frac{b}{a}$

Arctan Formula

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Some of the formulas of the arctan are as follows:
  1. Derivative of arctan is $\frac{\mathrm{d} \tan ^{-1}x}{\mathrm{d} x}$ = $\frac{1}{1 + x^2}$
  2. Integral of arctan is $\int \tan^{-1} x dx = x \tan^{-1} x - \frac{1}{2} \ln (1 + x^2) + C$.

Formula for Finding Angle in a Right Triangle:

Let $\theta$ be angle.

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

$\arctan (\frac{\text{Opposite side}}{\text{Adjacent side}}) = \theta$

Arctan Identities

  1. $\arctan x + \arctan y = \arctan(\frac{x + y}{1 - xy})$
  2. $\arctan x - \arctan y = \arctan(\frac{x - y}{1 + xy})$
  3. $\arctan x = \arcsin (\frac{x}{\sqrt{1 + x^2}})$

Arctan Properties

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Properties of arctan are as follow:

Negative Arguments of Arctan:

$\arctan (-x) = - \arctan (x)$

Reciprocal Arguments:

$\arctan (x) = arccot (\frac{1}{x})$

Composition of Arctan:

$\arctan (\tan x) = x$, for all $x$ in the range of arctan.

Some Important Values of Arctan:

$\arctan (1) = \frac{\pi}{4}$

$\arctan (0) = 0$

$\arctan (-1) = \frac{- \pi}{4}$

Graph of Arctan

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Arctan(x) is the set of all angles whose tan is x. The domain of the arctan function is all real numbers and the range is from $\frac{-\pi}{2}$ to $\frac{\pi}{2}$. Graph of arctan is represented as follows:

Graph for y = tan−1x


Domain of Arctan

Domain of arctan = R (All real numbers)

Range of Arctan

Range of arctan = ($\frac{-\pi}{2}$, $\frac{\pi}{2}$)

Arctan of 1

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Arctan of 1 is the angle whose tangent is equal to 1.

=> $\tan ^{-1} 1 = \tan ^{-1} (\tan \frac{\pi}{4})$

= $\frac{\pi}{4}$

[By using identity, $\tan ^{-1} (\tan x) = x$]

So, Arctan of 1 = $\frac{\pi}{4}$
Similarly, we can find the value of arctan 0.

Arctan 0

=> arctan 0 = arctan(tan 0)

[tan 0 = 0]

=> arctan(tan 0) = 0

[By using identity, arctan(tan x) = x]

So, arctan 0 = 0

Arctan of Infinity

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Arctan is undefined at infinity. However, as x approaches to infinity, arctan(x) tends to $\frac{\pi}{2}$.

=> As x -> $\infty$, then arctan x = $\frac{\pi}{2}$

Arctan Negative Infinity

The value of arctan can be found in the same way as the arctan. For x approaches to minus infinity, arctan(x) tends to - $\frac{\pi}{2}$.

=> As x -> -$\infty$, then arctan x = - $\frac{\pi}{2}$

Arctan Values

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Given below are some of the values of arctan for fixed points.
  1. $\arctan (1) = \frac{\pi}{4}$
  2. $\arctan (0) = 0$
  3. $\arctan (-1) = \frac{-\pi}{4}$

Hyperbolic Arctangent

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The inverse hyperbolic functions provide a hyperbolic angle corresponding to a given value of a hyperbolic function. It can be denoted as arctanh or $\tanh ^{-1}$. The hyperbolic arc-tangent is the inverse of the hyperbolic tangent function. If $y = a \tanh (x)$, then $x = \tanh ^{-1}$$\frac{y}{a}$

Arctan Series

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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

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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?
Opposite side = 5 cm

Adjacent side = 12 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?
Opposite side = 20 cm

Adjacent side = 25 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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