Inverse Trigonometric Functions - II

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Carefully understand the domain and range of following functions


4. f(x) = {\cot ^{ - 1}}x



Domain = ( - \infty ,\infty ) = R

Range = (0,\pi )

 You can observe from the above graph
{\cot ^{ - 1}}( - x) = \pi - {\cot ^{ - 1}}x, where x belongs to domain of arccot x



Comparison between the graph of arctan x and arccot x

You can observe from above graph,

 {\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \frac{\pi }{2},  x belongs to R i.e. domain of arctan(x) and arccot(x)




5. f(x) = {{{\rm cosec}\nolimits} ^{ - 1}}x

Domain = ( - \infty , - 1]\bigcup {[1,\infty ) = R - ( - 1,1)}

Range = \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right] - \{ 0\}


You can observe,

{\rm{cose}}{{\rm{c}}^{{\rm{ - 1}}}}{\rm{( - x) = - cose}}{{\rm{c}}^{{\rm{ - 1}}}}{\rm{x}}, where x belongs to domain of arccosec(x)




6. f(x) = se{c^{ - 1}}x



Domain = ( - \infty , - 1]\bigcup {[1,\infty ) = R - ( - 1,1)}

Range = [0,\pi ] - \left\{ {\frac{\pi }{2}} \right\}
From the above graph, you can observe

 {\sec ^{ - 1}}( - x) = \pi - {\sec ^{ - 1}}x, x belongs to domain of arcsec(x)




Comparison between the graph of arccosec x and arcsec x


Observation from above graph,
\cos e{c^{ - 1}}x + {\sec ^{ - 1}}x = \frac{\pi }{2}, where x belongs to domain of arccosec(x) or arcsec(x)

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