**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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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}$ Some of the formulas of the arctan are as follows:

### 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

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:**

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

$\arctan (0) = 0$

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

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**^{âˆ’1}x

### 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 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 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 = - $\frac{\pi}{2}$ Given below are some of the values of arctan for fixed points.

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.

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

** Solution: **

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

Then, $\theta = \arctan$$\frac{b}{a}$ Some of the formulas of the arctan are as follows:

- Derivative of arctan is $\frac{\mathrm{d} \tan ^{-1}x}{\mathrm{d} x}$ = $\frac{1}{1 + x^2}$
- Integral of arctan is $\int \tan^{-1} x dx = x \tan^{-1} x - \frac{1}{2} \ln (1 + x^2) + C$.

Let $\theta$ be angle.

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

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

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

Properties of arctan are as follow:

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

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

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

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

$\arctan (0) = 0$

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

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:

Domain of arctan = R (All real numbers)

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

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(tan 0)

[tan 0 = 0]

=> arctan(tan 0) = 0

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

So, arctan 0 = 0

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

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

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 = - $\frac{\pi}{2}$ Given below are some of the values of arctan for fixed points.

- $\arctan (1) = \frac{\pi}{4}$
- $\arctan (0) = 0$
- $\arctan (-1) = \frac{-\pi}{4}$

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.

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

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°.

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°.

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°.

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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