Question

If a line makes angles α, β, γ with positive directions of the coordinate axes, then the value of cos 2α + cos 2β + cos 2γ is __________.

Answer 1

Suppose a line makes angles α, β and γ with positive directions of the coordinate axes, then l = cosα, m = cosβ and n = cosγ
Here cos2α + cos2β + cos2γ
= 2cos²α – 1 + 2cos²β – 1 + 2cos²γ – 1
= 2(cos²α + cos²β + cos²γ) – 3
Since cos²α + cos²β + cos²γ
= l² + m² + n²
= 1
∴ cos2α + cos2β + cos2γ
= 2(cos²α + cos²β + cos²γ) – 3
= 2(1) – 3
i.e cos2α + cos2β + cos2γ = –1