Inverse Circular Functions: Domain & Range

Inverse circular functions, also known as inverse trigonometric functions, are the inverse operations of the trigonometric functions. They are used to determine the angle that corresponds to a given trigonometric ratio. In this article, we will explore the domain and range of the six primary inverse circular functions: arcsine ($\arcsin$), arccosine ($\arccos$), arctangent ($\arctan$), arccosecant ($\arccsc$), arcsecant ($\arcsec$), and arccotangent ($\arccot$).

Definitions and Notations

Before diving into the domain and range, let's define each inverse circular function and its notation:

• $\arcsin(x)$: Inverse sine function
• $\arccos(x)$: Inverse cosine function
• $\arctan(x)$: Inverse tangent function
• $\arccsc(x)$: Inverse cosecant function
• $\arcsec(x)$: Inverse secant function
• $\arccot(x)$: Inverse cotangent function

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For inverse circular functions, the domain and range are restricted to ensure that each function is a well-defined, single-valued function.

Here is a table summarizing the domain and range of each inverse circular function:

Function	Domain	Range
$\arcsin(x)$	$[-1, 1]$	$[-\frac{\pi}{2}, \frac{\pi}{2}]$
$\arccos(x)$	$[-1, 1]$	$[0, \pi]$
$\arctan(x)$	$(-\infty, \infty)$	$(-\frac{\pi}{2}, \frac{\pi}{2})$
$\arccsc(x)$	$(-\infty, -1] \cup [1, \infty)$	$[-\frac{\pi}{2}, \frac{\pi}{2}] \setminus {0}$
$\arcsec(x)$	$(-\infty, -1] \cup [1, \infty)$	$[0, \pi] \setminus {\frac{\pi}{2}}$
$\arccot(x)$	$(-\infty, \infty)$	$(0, \pi)$

Formulas and Examples

Arcsine ($\arcsin$)

The arcsine function is the inverse of the sine function. It takes a value between -1 and 1 and returns an angle in radians between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Example:

$$ \arcsin(0.5) = \frac{\pi}{6} $$

Arccosine ($\arccos$)

The arccosine function is the inverse of the cosine function. It takes a value between -1 and 1 and returns an angle in radians between $0$ and $\pi$.

Example:

$$ \arccos(0.5) = \frac{\pi}{3} $$

Arctangent ($\arctan$)

The arctangent function is the inverse of the tangent function. It can take any real number as an input and returns an angle in radians between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

Example:

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

Arccosecant ($\arccsc$)

The arccosecant function is the inverse of the cosecant function. Its domain excludes the interval $(-1, 1)$, and its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$ excluding $0$.

Example:

$$ \arccsc(2) = \frac{\pi}{6} $$

Arcsecant ($\arcsec$)

The arcsecant function is the inverse of the secant function. Similar to arccosecant, its domain excludes the interval $(-1, 1)$, and its range is $[0, \pi]$ excluding $\frac{\pi}{2}$.

Example:

$$ \arcsec(2) = \frac{\pi}{3} $$

Arccotangent ($\arccot$)

The arccotangent function is the inverse of the cotangent function. It can take any real number as an input and returns an angle in radians between $0$ and $\pi$.

Example:

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

Important Points to Remember

• The domains of the inverse circular functions are restricted to ensure that they are functions (i.e., each input has exactly one output).
• The ranges of the inverse circular functions are chosen to provide unique values for each input within the domain.
• The inverse circular functions are often used to solve trigonometric equations or to find the angles associated with given trigonometric ratios.
• When working with inverse circular functions, it is important to consider the principal values, which are the most commonly used values within the defined range of the function.

Understanding the domain and range of inverse circular functions is crucial for solving problems involving these functions, especially in calculus, geometry, and trigonometry.