Question

If cos $\alpha$, cos $\beta$, cos $\gamma$, are direction cosine of a line then value of $\sin^2 \, \alpha + \sin^2 \, \beta + \sin^2 \gamma$ is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

If a line OP makes angles $\alpha$, $\beta$, $\gamma$ respectively with x, y, z axes, the direction cosines are $\cos \alpha, \cos \beta, \cos \gamma$. Then, $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Consider $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ $= 1 - \cos^2 \alpha + 1 - \cos^2 \beta + 1 - \cos^2 \gamma$ $= 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma)$ $= 3 - 1 = 2$