Question

If $\alpha ,\beta$ and $\gamma$ are the angles which the vector $\overrightarrow{OP}$ (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct?
1. ${\cos }^2\alpha +{\cos }^2\beta ={\sin }^2\gamma$
2. ${\sin }^2\alpha +{\sin }^2\beta ={\cos }^2\gamma$
3. ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$
Select the correct answer using the code given below

• A. 1 and 2 only
• B. 2 and 3 only
• C. 1 and 3 only
• D. 1, 2 and 3

Answer 1

Answer: (C)

1 and 3 only

Given

$\alpha ,\beta ,\gamma$ are angles of coordinate axes

So dr's of $\overrightarrow{OP}$ are

$\cos \alpha ,\cos \beta ,\cos \gamma$

by property of dr's

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

${\cos }^2\alpha +{\cos }^2\beta =1-{\cos }^2\gamma$

By trigonometric identity

${\sin }^2\gamma +{\cos }^2\gamma =1$

${\cos }^2\gamma =1-{\sin }^2\gamma$

So

${\cos }^2\alpha +{\cos }^2\beta ={\sin }^2\gamma$

Hence statement 1 is correct

again by property of dr's

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

by using identity of trigonometry

$1-{\sin }^2\alpha +1-{\sin }^2\beta +1-{\sin }^2\gamma =1$

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

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

${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$

Hence statement 3 is correct.