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# Arctan

In trigonometry, there are six functions such as sin, cos, tan, cosec, sec and cot. There also exist inverse functions of each of those. Arctan is the inverse function of tangent function. It is also denoted by tan$^{-1}$. We can write that

If y is the tangent function of x i.e. $y$ = $tan\ x$ , then $x$ = $tan ^{-1} \ y$

$tan^{-1} \theta \neq $$\frac{1}{tan \theta} rather (tan \theta)^{-1}=$$\frac{1}{tan \theta}$

Tangent function possesses inverse when it has some limitation. When it is bounded between $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, we have inverse of tangent.

 Related Calculators Arctan Calculator

## Domain and Range

Domain of arctan = All real numbers

Range of arctan = [$-\frac{\pi}{2}, \frac{\pi}{2}$]

## Formulas Related to arctan

• arctan x = $\frac{\pi}{2}$$-\ arccot\ x\ =\ arctan\ ($$\frac{1}{x}$$) • arctan(- x) = - arctan(x) • tan(arctan x) = x • \frac{d}{dx} arctan x = \frac{1}{1+x^{2}} • \int arctan x dx = x arctan x - \frac{1}{2}$$ln(1 + x^{2})$ + c, where c is a constant.

## Examples

Example 1: Find the value of $arcsin(\frac{1}{2})$

Solution: Since $sin(\frac{-\pi}{6})$ = $\frac{-1}{2}$ for $\frac{-\pi}{2}$ $\leq$ $y$ $\leq$ $\frac{\pi}{2}$

Hence $arcsin(\frac{1}{2})$ = $\frac{-\pi}{6}$
Example 2: Find the value of $arcsin(\frac{\sqrt 3}{2})$

Solution: Since $sin(\frac{\pi}{3})$ = $\frac{\sqrt 3}{2}$ for $\frac{-\pi}{2}$ $\leq$ $y$ $\leq$ $\frac{\pi}{2}$

$arcsin(\frac{\sqrt 3}{2})$ = $\frac{\pi}{3}$

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