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A line makes $45^{\circ}$ with positive $\mathrm{x}$ -axis and makes equal angles with positive $y, z$ axes, respectively. What is the sum of the three angles which the line makes with positive $x, y$ and $z$ axes?
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Verified Answer
The correct answer is:
$165^{\circ}$
We know that sum of square of direction cosines $=1$ i.e. $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow \quad \cos ^{2} 45^{\circ}+\cos ^{2} \beta+\cos ^{2} \beta=1$
(As given $\alpha=45^{\circ}$ and $\beta=\gamma$ )
$\Rightarrow \quad \frac{1}{2}+2 \cos ^{2} \beta=1$
$\Rightarrow \cos ^{2} \beta=\frac{1}{4}$
$\Rightarrow \cos \beta=\pm \frac{1}{2}$, Negative value is discarded, since the line
makes angle with positive axes.
Hence, $\cos \beta=\frac{1}{2}$
$\Rightarrow \cos \beta=\cos 60^{\circ}$
$\Rightarrow \quad \beta=60^{\circ}$
Required sum $=\alpha+\beta+\gamma=45^{\circ}+60^{\circ}+60^{\circ}=165^{\circ}$
$\Rightarrow \quad \cos ^{2} 45^{\circ}+\cos ^{2} \beta+\cos ^{2} \beta=1$
(As given $\alpha=45^{\circ}$ and $\beta=\gamma$ )
$\Rightarrow \quad \frac{1}{2}+2 \cos ^{2} \beta=1$
$\Rightarrow \cos ^{2} \beta=\frac{1}{4}$
$\Rightarrow \cos \beta=\pm \frac{1}{2}$, Negative value is discarded, since the line
makes angle with positive axes.
Hence, $\cos \beta=\frac{1}{2}$
$\Rightarrow \cos \beta=\cos 60^{\circ}$
$\Rightarrow \quad \beta=60^{\circ}$
Required sum $=\alpha+\beta+\gamma=45^{\circ}+60^{\circ}+60^{\circ}=165^{\circ}$
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