Q193 / 270 · NDA 2017 (Phase 2) · Mathematics
If $\alpha, \beta$ and $\gamma$ are the angles which the vector $\overrightarrow{\mathrm{OP}}$ (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
- Correct answerC$1$ and $3$ only
- D$1,2$ and $3$
Solution
We know, $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$
$\Rightarrow \cos ^2 \alpha+\cos ^2 \beta=1-\cos ^2 \gamma=\sin ^2 \gamma$
$\therefore$ Statement 1 is correct.
Now, $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$
$\begin{aligned}
& \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \beta+1-\sin ^2 \gamma=1 \\& \Rightarrow 3-\left(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\right)=1 \\& \Rightarrow \sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=2\end{aligned}$
$\therefore$ Statement 3 is correct.
Hence, the correct answer is option: 'C'.