3.14 Inverse trigonometric functions

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

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

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 :

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

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

Therefore, we define arcsine function as :

$$f:[-\mathrm{1,1}]\to [-\frac{\pi }{2},\frac{\pi }{2}]\text{by f(x)}=\text{arcsin(x)}$$

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.

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

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

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 :

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

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

Therefore, we define arccosine function as :

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

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.

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

$$\text{Range of tangent}=R$$

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 :

$$\text{Domain of arctangent}=R$$

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

Therefore, we define arctangent function as :

$$f:R\to \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\text{by f(x)}=\text{arctan (x)}$$

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.

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

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

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 :

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

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

Therefore, we define arccosecant function as :