Question

If $\alpha, \beta, \gamma$ are the angles which a half ray makes with the positive direction of the axes, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

• A. $1$
• B. $2$
• C. $0$
• D. $-1$

Hint:

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ for direction cosines.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For direction cosines, $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.
Step 2: Detailed Explanation:
We know $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$. Then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = (1-\cos^2\alpha) + (1-\cos^2\beta) + (1-\cos^2\gamma) = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) = 3 - 1 = 2$.
Step 3: Final Answer:
$\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$.