# 3.14 Inverse trigonometric functions  (Page 2/4)

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$\text{Domain of sine}=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

$\text{Range of sine}=\left[-1,1\right]$

This redefinition renders sine function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

$\text{Domain of arcsine}=\left[-1,1\right]$

$\text{Range of arcsine}=\left[-\frac{\pi }{2,}\frac{\pi }{2}\right]$

Therefore, we define arcsine function as :

$f:\left[-1,1\right]\to \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arcsin(x)}$

The arcsin(x) .vs. x graph is shown here.

## Arccosine function

The arccosine function is inverse function of trigonometric cosine function. From the plot of cosine function, it is clear that an interval between 0 and $\pi$ includes all possible values of cosine function only once. Note that end points are included. The redefinition of domain of trigonometric function, however, does not change the range.

$\text{Domain of cosine}=\left[0,\pi \right]$

$\text{Range of cosine}=\left[-1,1\right]$

This redefinition renders cosine function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

$\text{Domain of arccosine}=\left[-1,1\right]$

$\text{Range of arccosine}=\left[0,\pi \right]$

Therefore, we define arccosine function as :

$f:\left[-1,1\right]\to \left[0,\pi \right]\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arccos(x)}$

The arccos (x) .vs. x graph is shown here.

## Arctangent function

The arctangent function is inverse function of trigonometric tangent function. From the plot of tangent function, it is clear that an interval between $-\pi /2$ and $\pi /2$ includes all possible values of tangent function only once. Note that end points are excluded. The redefinition of domain of trigonometric function, however, does not change the range.

$\text{Domain of tangent}=\left(-\frac{\pi }{2,}\frac{\pi }{2}\right)$

$\text{Range of tangent}=R$

This redefinition renders tangent function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

$\text{Domain of arctangent}=R$

$\text{Range of arctangent}=\left(-\pi /2,\pi /2\right)$

Therefore, we define arctangent function as :

$f:R\to \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arctan (x)}$

The arctan(x) .vs. x graph is shown here.

## Arccosecant function

The arccosecant function is inverse function of trigonometric cosecant function. From the plot of cosecant function, it is clear that union of two disjointed intervals between “ $-\pi /2$ and 0” and “0 and $\pi /2$ ” includes all possible values of cosecant function only once. Note that zero is excluded, but “ $-\pi /2$ “ and “ $\pi /2$ ” are included . The redefinition of domain of trigonometric function, however, does not change the range.

$\text{Domain of cosecant}=\left[-\pi /2,\pi /2\right]-\left\{0\right\}$

$\text{Range of cosecant}=\left(-\infty ,-1\right]\cup \left[1,\infty \right)=R-\left(-1,1\right)$

This redefinition renders cosecant function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

$\text{Domain of arccosecant}=R-\left(-1,1\right)$

$\text{Range of arccosecant}=\left[-\pi /2,\pi /2\right]-\left\{0\right\}$

Therefore, we define arccosecant function as :

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in general
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Period of sin^6 3x+ cos^6 3x