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Question
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
Options
0
1
6
12
Solution
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals 6.
Explanation:
We have cos–1α + cos–1β + cos–1γ = 3π
⇒ cos–1α + cos–1β + cos–1γ = π + π + π
⇒ cos–1α = π, cos–1β = π and cos–1γ = π
⇒ α = cos π, β = cos π and γ = cos π
∴ α = – 1, β = – 1 and γ = – 1
Which gives α = β = γ = –1
So α(β + γ) + β(γ + α) + γ(α + β)
⇒ (– 1)(– 1 – 1) + (– 1)(– 1 – 1) + (– 1)(– 1 – 1)
⇒ (– 1)(– 2) + (– 1)(– 2) + (– 1)(– 2)
⇒ 2 + 2 + 2
⇒ 6
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