First slide

Introduction to ITF

Question

$If \cos^{- 1} α + \cos^{- 1} β + \cos^{- 1} γ = 3 π, then α (β + γ) + β (γ + α) + γ (α + β) equals$

Moderate

A

0
B

12
C

1
D

6

Solution

$\begin{array}{l} We have, \cos^{- 1} α + \cos^{- 1} β + \cos^{- 1} γ = 3 π \\ We know that, 0 \leq \cos^{- 1} x \leq π \\ \Rightarrow \cos^{- 1} α + \cos^{- 1} β + \cos^{- 1} γ = 3 π is possible if and only if \\ \cos^{- 1} α = \cos^{- 1} β = \cos^{- 1} γ = π \\ \Rightarrow α = β = γ = - 1 \\ ∴ α (β + γ) + β (γ + α) + γ (α + β) \\ = - 1 (- 1 - 1) - 1 (- 1 - 1) - 1 (- 1 - 1) = 6 \end{array}$

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