Formulas and Tables

Trigonometry

Inverse Trigonometric Functions

Argument of an inverse function: \(x\)
Angle (value of an inverse function): \(y\)

Set of real numbers: \(\mathbb{R}\)
Real number: \(a\)

The inverse trigonometric functions include the following \(6\) functions: arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant.
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.
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 \(y \in \left[ { – \large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right]\).

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 \(y \in \left[ {0,\pi} \right]\).

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 \(\left( {-\large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right).\)

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 \(y \in \left[ {0,\pi} \right]\).

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 \(y \in \) \(\left[ {0,\large{\frac{\pi}{2}}\normalsize} \right) \cup\) \(\left( {\large{\frac{\pi}{2}}\normalsize,\pi } \right]\).

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 \(y \in \) \(\left[ { – \large{\frac{\pi}{2}}\normalsize,0} \right) \cup \left( {0,\large{\frac{\pi}{2}}\normalsize} \right]\).