3.14 Inverse trigonometric functions

$$f:R-\left(-\mathrm{1,}1\right)\to [-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}\text{by f(x)}=\text{arccosec (x)}$$

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

Arcsecant function

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

$$\text{Domain of secant}=[\mathrm{0,}\pi /2)\cup (\pi /\mathrm{2,}\pi ]=\left[\mathrm{0,}\pi \right]-\{\pi /2\}$$

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

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

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

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

Therefore, we define arcsecant function as :

$$f:R-\left(-\mathrm{1,}1\right)\to \left[\mathrm{0,}\pi \right]-\{\pi /2\}\text{by f(x)}=\text{arcsec (x)}$$

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

Arccotangent function

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

$$\text{Domain of cotangent}=\left(\mathrm{0,}\pi \right)$$

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

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

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

$$\text{Range of arccotangent}=\left(\mathrm{0,}\pi \right)$$

Therefore, we define arccotangent function as :

$$f:R\to \left(\mathrm{0,}\pi \right)\text{by f(x)}=\text{arccot (x)}$$

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

Example