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Geometry Level 4

Find the range of the function f ( x ) = sin 1 x + cos 1 x + tan 1 x f(x)=\sin^{-1}x+\cos^{-1}x+\tan^{-1}x .

[ 0 , π ] [0,\pi ] [ π 4 , 3 π 4 ] \left[\frac{\pi}{4},\frac{3\pi}{4}\right] ( π 4 , 3 π 4 ) \left(\frac{\pi}{4},\frac{3\pi}{4}\right) None of these choices ( 0 , π ) (0,\pi)

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1 solution

Rishabh Jain
May 14, 2016

Since sin 1 \sin^{-1} and cos 1 \cos^{-1} are defined for x [ 1 , 1 ] x\in[-1,1] , hence domain of f ( x ) f(x) is [ 1 , 1 ] [-1,1] and using sin 1 x + cos 1 x = π 2 \sin^{-1}x+\cos^{-1}x=\dfrac{\pi}{2} :- f ( x ) = π 2 + tan 1 x , x [ 1 , 1 ] \large f(x)=\dfrac{\pi}{2}+\tan^{-1}x,~~\small{ x\in[-1,1]}

And since tan 1 x \tan^{-1}x is strictly increasing in [ 1 , 1 ] [-1,1] \therefore

π 2 + tan 1 ( 1 ) f ( x ) π 2 + tan 1 1 \large \dfrac{\pi}{2}+\tan^{-1}(-1)\leqslant f(x)\leqslant \dfrac{\pi}{2}+\tan^{-1}1 Hence,

π 4 f ( x ) 3 π 4 \Large\color{#EC7300}{\boxed{\color{#20A900}{\dfrac{\pi}{4}\leqslant f(x)\leqslant \dfrac{3\pi}{4}}}}

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