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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?
$1.$ $\quad \cos ^{2} \alpha+\cos ^{2} \beta=\sin ^{2} \gamma$
$2.$ $\quad \sin ^{2} \alpha+\sin ^{2} \beta=\cos ^{2} \gamma$
$3.$ $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$
Select the correct answer using the code given below.
Options:
$1.$ $\quad \cos ^{2} \alpha+\cos ^{2} \beta=\sin ^{2} \gamma$
$2.$ $\quad \sin ^{2} \alpha+\sin ^{2} \beta=\cos ^{2} \gamma$
$3.$ $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$
Select the correct answer using the code given below.
Solution:
2863 Upvotes
Verified Answer
The correct answer is:
1 and 3 only
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, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow 1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow 3-\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\right)=1 \Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma$
$=2 .$
$\therefore$ Statement 3 is correct.
$\Rightarrow \cos ^{2} \alpha+\cos ^{2} \beta=1-\cos ^{2} \gamma=\sin ^{2} \gamma$
$\therefore$ Statement 1 is correct.
Now, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow 1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow 3-\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\right)=1 \Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma$
$=2 .$
$\therefore$ Statement 3 is correct.
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