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KCET 2025 Math Paper

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Question : 38 of 60

Marks: +1, -0

If a line makes angles

90^{\circ}, 60^{\circ}

\theta

x, y

and z axes respectively, where

\theta

is acute, then the value of

\theta

is

$\frac{\pi}{6}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{2}$

Solution:

A line's direction cosines along the

x, y, z

axes are

\cos \alpha, \cos \beta, \cos \gamma

where

\alpha=90^{\circ}, \beta=60^{\circ}

, and

\gamma=\theta

. They satisfy

\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1

Compute the known cosines:

\cdot \cos \alpha = \cos 90^{\circ} = 0

\cdot \cos \beta = \cos 60^{\circ} = \frac{1}{2}

Plug in and solve for

\cos \gamma

:

0^2 + \left(\;\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \;\; \Rightarrow \;\; \;\frac{1}{4} + \cos^2 \theta = 1 \;\; \Rightarrow \;\; \cos^2 \theta = \;\frac{3}{4}

Since

\theta

is acute,

\cos \theta > 0

, so

\cos \theta = \frac{\sqrt{3}}{2} .

Therefore

\theta = \arccos \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6}

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