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: \(a\)

  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 \(y \in \left[ { – \large{\frac{\pi}{2}}\normalsize, \large{\frac{\pi}{2}}\normalsize} \right]\).
Inverse of the sine (arcsine)
  1. 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 cosine (arccosine)
  1. 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 tangent (arctangent)
  1. 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 cotangent (arccotangent)
  1. 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 secant (arcsecant)
  1. 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 \(y \in \) \(\left[ { – \large{\frac{\pi}{2}}\normalsize,0} \right) \cup \left( {0,\large{\frac{\pi}{2}}\normalsize} \right]\).
Inverse of the cosecant (arccosecant)
  1. Principal values of the arcsine and arccosine functions (in degrees)
Principal values of the arcsine and arccosine functions
  1. Principal values of the arctangent and arccotangent functions (in degrees)
Principal values of the arctangent and arccotangent functions