If α+β+γ=π then the minimum value ofcosA+cosB+cosCis zerois positivelies between −2 and −3is −3

The given equation is α+β+γ=π. We know that cos(π - θ) = -cosθ. So, we can write cosα + cosβ + cosγ as cos(π - β - γ) + cos(π - α - γ) + cos(π - α - β).

This simplifies to -cos(β + γ) - cos(α + γ) - cos(α + β).

Now, we know that -2 ≤ cosθ ≤ 2 for any real number θ. So, the minimum value of cos(β + γ) + cos(α + γ) + cos(α + β) is -2 - 2 - 2 = -6.

But, we have a negative sign in front of the whole expression, so the minimum value of cosα + cosβ + cosγ is -(-6) = 6.

Therefore, the minimum value of cosA + cosB + cosC is 6.