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Inverse Trigonometric Functions – Summary
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The domains and ranges (principal value branches) of inverse trigonometric functions are given in the following table:
Functions
Domain
Range
(Principal Value Branches)
(Principal Value Branches)
{y = \sin^{-1} x}
[1, 1]
{\left[-\dfrac{π}{2}, \dfrac{π}{2}\right]}
{y = \cos^{-1} x}
[1, 1]
[0, π]
{y = \cosec^{-1} x}
R – (1, 1)
{\left[-\dfrac{π}{2}, \dfrac{π}{2}\right] - \{0\}}
{y = \sec^{-1} x}
R – (-1, 1)
{[0, π] - \left\{\dfrac{π}{2}\right\}}
{y = \tan^{-1} x}
R
{\left(-\dfrac{π}{2}, \dfrac{π}{2}\right)}
{y = \cot^{-1} x}
R
(0, π)
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Both {\sin^{-1} x} and {(\sin x)^{-1}} are different functions. It is important that we do not confuse {\sin^{-1} x} as {(\sin x)^{-1}}. In fact
{(\sin x)^{-1} = \dfrac{1}{\sin x}}
This is true for all other trigonometric functions.
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The value of an inverse trigonometric function which lies in its principal value branch is called the principal value of that inverse trigonometric functions.
For suitable values of domain we have
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{y = \sin^{-1} x}
⇒ {x = \sin y}
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{x = \sin y}
⇒ {y = \sin^{-1} x}
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{\sin (\sin^{-1} x) = x}
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{\sin^{-1} (\sin x) = x}
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{\sin^{-1} \left(\dfrac{1}{x}\right) = \cosec^{-1} x}
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{\cos^{-1} \left(\dfrac{1}{x}\right) = \sec^{-1} x}
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{\tan^{-1} \left(\dfrac{1}{x}\right) = \cot^{-1} x}
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{\cos^{-1} (-x) = π - \cos^{-1} x}
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{\sec^{-1} (-x) = π - \sec^{-1} x}
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{\cot^{-1} (-x) = π - \cot^{-1} x}
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{\sin^{-1} (-x) = -\sin^{-1} x}
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{\cosec^{-1} (-x) = -\cosec^{-1} x}
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{\tan^{-1} (-x) = -\tan^{-1} x}
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{\sin^{-1} x + \cos^{-1} x = \dfrac{π}{2}}
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{\cosec^{-1} x + \sec^{-1} x = \dfrac{π}{2}}
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{\tan^{-1} x + \cot^{-1} x = \dfrac{π}{2}}
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{\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\dfrac{x + y}{1 - xy}\right)}, {xy \lt 1}
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{\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\dfrac{x + y}{1 - xy}\right)}, {xy \gt 1}; {x, y \gt 0}
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{\tan^{-1} x - \tan^{-1} y = \tan^{-1} \left(\dfrac{x - y}{1 + xy}\right)}, {xy \gt -1}
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{2\tan^{-1} x}
=
{\sin^{-1} \dfrac{2x}{1 + x^2}}
=
{\cos^{-1} \dfrac{1 - x^2}{1 + x^2}}
=
{\tan^{-1} \dfrac{2x}{1 - x^2}}
