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Consider the following relations among the angles $\alpha, \beta$ and $\gamma$ made by a vector with the coordinate axes
I. $\quad \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
II. $\quad \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$
Which of the above is/are correct?
Options:
I. $\quad \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
II. $\quad \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$
Which of the above is/are correct?
Solution:
2037 Upvotes
Verified Answer
The correct answer is:
Only I
We have, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow 2 \cos ^{2} \alpha+2 \cos ^{2} \beta+2 \cos ^{2} \gamma=2$
$\Rightarrow 2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1=2-3$
$\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
Hence statement $-\mathrm{I}$ is correct.
and now from (i), $1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$
Hence, only statement I is correct.
$\Rightarrow 2 \cos ^{2} \alpha+2 \cos ^{2} \beta+2 \cos ^{2} \gamma=2$
$\Rightarrow 2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1=2-3$
$\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
Hence statement $-\mathrm{I}$ is correct.
and now from (i), $1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$
Hence, only statement I is correct.
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