A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2 β = 3 sin2 θ, then cos2 θ equals

Question

A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2 β = 3 sin2 θ, then cos2 θ equals (A) (2/5) (B) (1/5) (C) (3/5) (D) (2/3)

Tardigrade Admin · Accepted Answer

The direction cosines of the line are cos θ , cos β, cosθ ∴ cos2 θ + cos2 β+ cos2 θ =1 ⇒ 2 cos2 θ = sin2 β = 3 sin2 θ (given) ⇒ 2 cos2 θ = 3-3 cos2 θ ∴ cos2 θ = ( 3/5)

Q. A line makes the same angle $θ$ , with each of the x and z axis. 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

$\frac{2}{5}$

$\frac{1}{5}$

$\frac{3}{5}$

$\frac{2}{3}$

The direction cosines of the line are $cos θ, cos β, cos θ$
$∴ cos^{2} θ + cos^{2} β + cos^{2} θ = 1$
$\Rightarrow 2 cos^{2} θ = sin^{2} β = 3 sin^{2} θ$ (given)
$\Rightarrow 2 cos^{2} θ = 3 - 3 cos^{2} θ$
$∴ cos^{2} θ = \frac{3}{5}$

Q. A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2β=3sin2θ, then cos2θ equals

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