Maths · Inverse trigonometrical functions and their properties
If =1, \) where [.] denotes the greatest integer function, the lies in the inter
If \( \left[\sin ^{-1} \cos ^{-1} \sin ^{-1} \tan ^{-1} \theta\right]=1, \) where [.] denotes the greatest integer function, the \( \theta \) lies in the interval
- A. [tan sin cos \( 1, \text { sin tan } \cos \sin 1] \)
- B. [sin tan cos \( 1, \text { tan } \sin \cos \sin 1] \)
- C. \( [\tan \sin \cos 1, \tan \sin \cos \sin 1] \)
- D. None of these
Step-by-step solution
Let w = sin^{-1}(cos^{-1}(sin^{-1}(tan^{-1}θ))). Then [w] = 1 implies 1 ≤ w < 2. Since sin^{-1} ranges in [-π/2, π/2], we need sin^{-1}(c) ∈ [1, π/2) ⇒ c ∈ [sin1, 1] where c = cos^{-1}(b). Thus cos^{-1}(b) ∈ [sin1, 1] ⇒ b ∈ [cos1, cos(sin1)]. b = sin^{-1}(a) ⇒ a ∈ [sin(cos1), sin(cos(sin1))]. a = tan^{-1}θ ⇒ θ ∈ [tan(sin(cos1)), tan(sin(cos(sin1)))], which matches option C.
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