Question

A line makes the same angle \(\alpha\) with each of the \(X\) and \(Z\) axes. If the angle \(\beta\) which it makes with the \(Y\)-axis is such that \(\sin ^{2}\beta =2\sin ^{2}\alpha\), then the value of \(\cos ^{2}\alpha\) is ...

• A. \(\frac{1}{2}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{3}{5}\)
• D. \(\frac{4}{5}\)

Hint:

Use that the sum of squares of the direction cosines is 1 together with the given relation.

Answer 1

The Correct Option is A

Step 1: Write in terms of sin squared
$\sin^{2}\beta=2\sin^{2}\alpha$, and $\cos^{2}\alpha+\cos^{2}\alpha+\cos^{2}\beta=1$ gives $\sin^{2}\beta=2\cos^{2}\alpha$.

Step 2: Equate
So $2\sin^{2}\alpha=2\cos^{2}\alpha$, i.e. $\sin^{2}\alpha=\cos^{2}\alpha$.

Step 3: Solve
Using $\sin^{2}\alpha+\cos^{2}\alpha=1$, we get $\cos^{2}\alpha=\frac{1}{2}$.

Final Answer:
Hence cos squared alpha equals one half. \[ \boxed{\text{(A) }\dfrac{1}{2}} \]