A line makes the same angle θ with each of the x and z axes. If the angle β, which it makes with y -axis, is such that ^2β=3^2θ, then ^2θ equals - Mathematics MCQ AIEEE 2004 2026 | ExamTest.live

hardMCQAIEEE 20042026Mathematics

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A line makes the same angle θ with each of the x and z axes. If the angle β, which it makes with y -axis, is such that sin^{2} β = 3 sin^{2} θ, then cos^{2} θ equals

Solution & Step-by-step Explanation

Let the direction cosines be l = cos θ, m = cos β, n = cos θ . We know l^{2} + m^{2} + n^{2} = 1 \Rightarrow cos^{2} θ + cos^{2} β + cos^{2} θ = 1 . 2 cos^{2} θ + cos^{2} β = 1 cos^{2} β = 1 - 2 cos^{2} θ . Using sin^{2} β = 1 - cos^{2} β : sin^{2} β = 1 - (1 - 2 cos^{2} θ) = 2 cos^{2} θ . The given condition is sin^{2} β = 3 sin^{2} θ . 2 cos^{2} θ = 3 sin^{2} θ = 3 (1 - cos^{2} θ) 2 cos^{2} θ = 3 - 3 cos^{2} θ 5 cos^{2} θ = 3 \Rightarrow cos^{2} θ = 3/5 .

Try it yourself before checking the explanation above.

A line makes the same angle θ with each of the x and z axes. If the angle β, which it makes with y -axis, is such that sin^{2} β = 3 sin^{2} θ, then cos^{2} θ equals

\frac{2}{3}

\frac{1}{5}

\frac{3}{5}

\frac{2}{5}

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