If a line makes angles $α, β, γ$ with positive direction co-ordinate axes then prove that ${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$

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Solution

The angles made by line with axis is $α, β, γ$
For a line ${cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = 1 1 - {sin}^{2} α + 1 - {sin}^{2} β + 1 - {sin}^{2} γ = 1 3 - ({sin}^{2} α + {sin}^{2} β + {sin}^{2} γ) = 1 {sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$

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[RPET 2000; AMU 2002; MP PET 1989, 98, 2000, 03, Pb. CET 2001]

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If a line makes angles α,β,γ with positive direction co-ordinate axes then prove that sin2α+sin2β+sin2γ=2

The angles made by line with axis is α,β,γFor a line cos2α+cos2β+cos2γ=11−sin2α+1−sin2β+1−sin2γ=13−(sin2α+sin2β+sin2γ)=1sin2α+sin2β+sin2γ=2

If a line makes angles $α, β, γ$ with positive direction co-ordinate axes then prove that ${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$

The angles made by line with axis is $α, β, γ$
For a line ${cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = 1 1 - {sin}^{2} α + 1 - {sin}^{2} β + 1 - {sin}^{2} γ = 1 3 - ({sin}^{2} α + {sin}^{2} β + {sin}^{2} γ) = 1 {sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$