Question:
A line makes the same angle $\theta$ with the x and z-axis. If the angle $\beta$ which it makes with the y-axis is such that $\sin^2\beta = 3\sin^2\theta$ then find the value of $\cos^2\theta$.
As I know that the angle made with x and z-axis is the same, I can assume the line to be parallel to a vector: $a\hat i + b\hat j + a\hat k$
So I know that: $$\cos \theta = \frac{a}{\sqrt{2a^2 + b^2}}$$ $$\cos \beta = \frac{b}{\sqrt{2a^2 + b^2}}$$
When I attempt to solve this (using the relation of the sines given), I get: $$3b^2 = -a^2$$ Which doesn't seem to be right. Any hints?
If 'a line makes the same angle $\theta$ with the x and z-axis' then $\theta$ is at least a half of a right angle (when the line lies in $OXZ$ plane as a bisector of the first quarter), then its sine is at least $\sqrt 2/2$ and $$3\sin^2\theta \ge 1.5$$
And no $\beta$ satisfies $$\sin^2\beta \ge 1.5$$