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)

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 ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma$
$=1-{\cos }^2\alpha +1-{\cos }^2\beta +1-{\cos }^2\gamma$
$=3-({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )$
$=3-1=2$