If a line angles α, β, γ with positive directions of the coordinate axes, then the value of sin²α + sin²β + sin² γ is __________.

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Solution

Given, a line makes angles α, β and γ with positive directions of x-axis
Then l = cosα, m = cosβ and n = cosγ
Here sin²α + sin²β + sin²γ
= 1 – cos²α + 1 – cos²β + 1 – cos²γ
= 3 – (cos²α + cos²β + cos²γ)
= 3 – (l² + m² + n²)
= 3 – 1
i.e sin²α + sin²β + sin²γ = 2

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