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Inverse Trigonometry

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Up nextPrincipal values of inverse functions $\sin^{-1}$ in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ , $\cos^{-1}$ in $[0, \pi]$ , $\tan^{-1}$ in $(-\frac{\pi}{2}, \frac{\pi}{2})$ : one angle, in that rangeStart

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Inverse trigonometric functions0 / 4

Principal values of inverse functions $\sin^{-1}$ in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ ,

\cos^{-1}

in

[0, \pi]

,

\tan^{-1}

in

(-\frac{\pi}{2}, \frac{\pi}{2})

:

one angle, in that range

Start

sin⁻¹(sin x) and its relatives

\sin^{-1}(\sin x) = x

only for

x

in the principal range; otherwise find the angle in the range with the same sineLocked

Properties of inverse functions

\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}

;

\tan^{-1}x + \tan^{-1}y = \tan^{-1}\dfrac{x + y}{1 - xy}

when

xy < 1

Locked

Equations in inverse functionsCombine the inverse terms with the properties, take the ratio of both sides, then keep only roots in the principal rangesLocked