Question

A line makes the same angle $\theta$ with each of the $xandz$- axes. If it makes the angle $\beta$with $y$-axis such that

${\sin }^2\beta =3{\sin }^2\theta ,$ then ${\cos }^2\theta$ equals.

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

Answer 1

The correct option is A

$\frac{3}{5}$

Explanation for the correct options:

Finding the value of ${\mathbf{cos}}^{\mathbf{2}}\mathbf{\theta }$:

Since it is given that the line makes an angle $\theta$ with the $xandz$ axes, and angle $\beta$with $y-$axis,

$\begin{array}{rcl}{\cos }^2\theta +{\cos }^2\theta +{\cos }^2\beta & =& 1\\ 2{\cos }^2\theta +{\cos }^2\beta & =& 1\\ 2(1-{\sin }^2\theta )+1-{\sin }^2\beta & =& 1\\ 2(1-{\sin }^2\theta )+1-3{\sin }^2\theta & =& 1\mathbf{[}\mathbf{\because }{\mathbf{sin}}^{\mathbf{2}}\mathbf{\beta }\mathbf{=}\mathbf{3}{\mathbf{sin}}^{\mathbf{2}}\mathbf{\theta }\mathbf{]}\\ 2-2{\sin }^2\theta +1-3{\sin }^2\theta & =& 1\\ 3-5{\sin }^2\theta & =& 1\\ {\sin }^2\theta & =& \frac{2}{5}\\ 1-{\cos }^2\theta & =& \frac{2}{5}\\ {\cos }^2\theta & =& 1-\frac{2}{5}\\ {\cos }^2\theta & =& \frac{3}{5}\end{array}$

Hence, the correct answer is option (A)