First slide

Properties of ITF

Question

If $[\sin^{- 1} \cos^{- 1} \sin^{- 1} \tan^{- 1} θ] = 1$ , where [.] denotes the greatest integer function, the θ lies in the interval

Moderate

A

[tan sin cos1, sin tan cos sin1]
B

[sin tan cos1, tan sin cos sin1]
C

[tan sin cos1, tan sin cos sin1]
D

none of these

Solution

We have, $[\sin^{- 1} \cos^{- 1} \sin^{- 1} \tan^{- 1} θ] = 1$
$\begin{array}{l} \Rightarrow 1 \leq \sin^{- 1} \cos^{- 1} \sin^{- 1} \tan^{- 1} θ \leq \frac{π}{2} \\ \Rightarrow \sin 1 \leq \cos^{- 1} \sin^{- 1} \tan^{- 1} θ \leq 1 \\ \Rightarrow \cos \sin 1 \geq \sin^{- 1} \tan^{- 1} θ \geq \cos 1 \\ \Rightarrow \sin \cos \sin 1 \geq \tan^{- 1} θ \geq \sin \cos 1 \\ \Rightarrow \tan \sin \cos \sin 1 \geq θ \geq \tan \sin \cos 1 \end{array}$

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