# Formulas

## Trig Identities # Inverse Trigonometric Functions

Argument of an inverse function: $$x$$
Angle (value of an inverse function): $$y$$
Set of real numbers: $$\mathbb{R}$$
Real number: $$a$$
1. The inverse trigonometric functions include the following $$6$$ functions: arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant.
2. Because the original trigonometric functions are periodic, the inverse functions are, generally speaking, multivalued. To ensure a one-to-one matching between the two variables, the domains of the original trigonometric functions may be restricted to their principal branches. For example, the function $$y = \sin x$$ is considered only on the interval $$x \in \left[ { – \large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right]$$. On this interval, the inverse of the sine function is uniquely determined.
3. Inverse of the sine (arcsine)
The arcsine of a number $$a$$ (denoted by $$\arcsin a$$) is the value of the angle $$x$$ in the interval $$x \in \left[ { – \large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right],$$ at which $$\sin x = a.$$ The inverse function $$y = \arcsin x$$ is defined for $$x \in \left[ { -1,1} \right],$$ its range is $$y \in \left[ { – \large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right]$$.
4. Inverse of the cosine (arccosine)
The arccosine of a number $$a$$ (denoted by arccos $$\arccos a$$) is the value of the angle $$x$$ in the interval $$\left[ {0,\pi} \right]$$, at which $$\cos x = a$$. The inverse function $$y = \arccos x$$ is defined for $$x \in \left[ { -1,1} \right]$$, its range is $$y \in \left[ {0,\pi} \right]$$.
5. Inverse of the tangent (arctangent)
The arctangent of a number a (denoted by $$\arctan a$$) is the value of the angle $$x$$ in the open interval $$\left( {-\large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right),$$ at which $$\tan x = a$$. The inverse function $$y = \arctan x$$ is defined for all $$x \in \mathbb{R}$$, the range of the arctangent is $$\left( {-\large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right).$$
6. Inverse of the cotangent (arccotangent)
The arccotangent of a number $$a$$ (denoted by $$\text{arccot } a$$) is the value of the angle $$x$$ in the open interval $$\left[ {0,\pi} \right],$$ at which $$\cot x = a$$. The inverse function $$y = \text{arccot } x$$ is defined for all $$x \in \mathbb{R}$$, its range is $$y \in \left[ {0,\pi} \right]$$.
7. Inverse of the secant (arcsecant)
The arcsecant of a number $$a$$ (denoted by $$\text{arcsec } a$$) is the value of the angle $$x,$$ at which $$\sec x = a$$. The inverse function $$y = \text{arcsec } x$$ is defined for $$x \in$$ $$\left( { – \infty , – 1} \right] \cup$$ $$\left[ {1,\infty } \right)$$, its range is the set $$y \in$$ $$\left[ {0,\large{\frac{\pi}{2}}\normalsize} \right) \cup$$ $$\left( {\large{\frac{\pi}{2}}\normalsize,\pi } \right]$$.
8. Inverse of the cosecant (arccosecant)
The arccosecant of a number $$a$$ (denoted by $$\text{arccsc } a$$) is the value of the angle $$x,$$ at which $$\csc x = a$$. The inverse function $$y = \text{arccsc } x$$ is defined for $$x \in$$ $$\left( { – \infty , – 1} \right] \cup \left[ {1,\infty } \right)$$, its range is the set $$y \in$$ $$\left[ { – \large{\frac{\pi}{2}}\normalsize,0} \right) \cup \left( {0,\large{\frac{\pi}{2}}\normalsize} \right]$$.
9. Principal values of the arcsine and arccosine functions (in degrees)
10. Principal values of the arctangent and arccotangent functions (in degrees)