Precalculus

Trigonometry

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Inverse Trigonometric Functions

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 sine function is considered only on the interval \(\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right].\) On this interval, the inverse of the sine function is uniquely determined. Similarly, the domains are also restricted for other trigonometric functions.

Inverse of the Sine (Arcsine)

The arcsine of a number \(x\) (denoted by \(\arcsin x\)) is the value of the angle \(y\) in the interval \(\left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\) at which \(\sin y = x.\)

The inverse function \(y = \arcsin x\) is defined for \(x \in \left[ { -1,1} \right],\) its range is \(y \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right].\)

The arcsine function is odd:

\[\arcsin \left( { – x} \right) = – \arcsin x.\]

The graph of \(y = \arcsin x\) is shown below:

Graph of the inverse sine function
Figure 1.

Inverse of the Cosine (Arccosine)

The arccosine of a number \(x\) (denoted by \(\arccos x\)) is the value of the angle \(y\) in the interval \(\left[ {0,\pi} \right]\) at which \(\cos y = x\).

The inverse function \(y = \arccos x\) is defined for \(x \in \left[ { -1,1} \right]\), its range is \(y \in \left[ {0,\pi} \right]\).

The arccosine function is neither odd nor even:

\[\arccos \left( { – x} \right) = \pi – \arccos x.\]

The arccosine graph is given in Figure \(2.\)

Graph of the inverse cosine function
Figure 2.

Inverse of the Tangent (Arctangent)

The arctangent of a number \(x\) (denoted by \(\arctan x\)) is the angle \(y\) in the open interval \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\) such that \(\tan y = x.\)

The inverse function \(y = \arctan x\) is defined for all \(x \in \mathbb{R}\), the range of the arctangent is \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right).\)

The function \(y = \arctan x\) is odd:

\[\arctan \left( { – x} \right) = – \arctan x.\]
Graph of the inverse tangent function
Figure 3.

Inverse of the Cotangent (Arccotangent)

The arccotangent of a number \(x\) (denoted by \(\text{arccot } x\)) is the value of the angle \(y\) in the open interval \(\left( {0,\pi} \right)\) at which \(\cot y = x.\)

The inverse function \(y = \text{arccot } x\) is defined for all \(x \in \mathbb{R}\), its range is \(y \in \left( {0,\pi} \right).\)

The arccotangent function is neither odd nor even:

\[\text{arccot}\left( { – x} \right) = \pi – \text{arccot }x.\]
Graph of the inverse cotangent function
Figure 4.

Inverse of the Secant (Arcsecant)

The arcsecant of a number \(x\) (denoted by \(\text{arcsec } x\)) is the value of the angle \(y\) such that \(\sec y = x.\)

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

The arcsecant function is neither odd nor even. The negative angle identity for arcsecant has the form:

\[\text{arcsec}\left( { – x} \right) = \pi – \text{arcsec }x.\]
Graph of the inverse secant function
Figure 5.

Inverse of the Cosecant (Arccosecant)

The arccosecant of a number \(x,\) denoted \(\text{arccsc } x,\) is the value of the angle \(y\) at which \(\csc y = x.\)

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

The arccosecant function is odd:

\[\text{arccsc} \left( { – x} \right) = – \text{arccsc } x.\]

The graph of \(y = \text{arccsc } x\) looks as follows:

Graph of the inverse cosecant function
Figure 6.

See solved problems on Page 2.

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