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

Answer 1

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