Question

A line makes the same angle $\theta$ with each of the X and Z-axes. If the angle $\beta$ which it makes with Y-axis is such that $\sin^2 \beta = 3 \sin^2 \theta$, then $\cos^2 \theta$ equals

• A. $\frac{2}{5}$
• B. $\frac{1}{5}$
• C. $\frac{3}{5}$
• D. $\frac{2}{3}$

Hint:

To solve such problems, start by using the Pythagorean identity and relate the unknown angles through algebraic equations.

Answer 1

The Correct Option is C

Given, $\sin^2 \beta = 3 \sin^2 \theta$, and the line makes equal angles with the X and Z axes.
This implies that $\cos^2 \theta$ for the line will satisfy the equation: \[ \sin^2 \beta + 2 \cos^2 \theta = 1 \] Substitute the value of $\sin^2 \beta$ and solve: \[ 3 \sin^2 \theta + 2 \cos^2 \theta = 1 \] Since $\sin^2 \theta + \cos^2 \theta = 1$, we substitute $\sin^2 \theta = 1 - \cos^2 \theta$ into the equation, and solving the equation gives $\cos^2 \theta = \frac{3}{5}$.