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A line making angles $45^{\circ}$ and $60^{\circ}$ with the positive directions of the axes of $x$ and $y$ makes with the positive direction of $z$-axis, an angle of
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Verified Answer
The correct answer is:
$60^{\circ}$
Let $\gamma$ is the required angle, then
$$
\begin{array}{l}
\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} \gamma=1 \\
\Rightarrow \frac{1}{2}+\frac{1}{4}+\cos ^{2} \gamma=1 \\
\Rightarrow \cos ^{2} \gamma=1-\frac{3}{4}=\frac{1}{4} \\
\Rightarrow \cos \gamma=\frac{1}{2} \Rightarrow \gamma=60^{\circ} \\
\end{array}
$$
$$
\begin{aligned}
\Rightarrow & \cos \gamma=\frac{1}{2} \Rightarrow \gamma=60^{\circ}
\end{aligned}
$$
$$
\begin{array}{l}
\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} \gamma=1 \\
\Rightarrow \frac{1}{2}+\frac{1}{4}+\cos ^{2} \gamma=1 \\
\Rightarrow \cos ^{2} \gamma=1-\frac{3}{4}=\frac{1}{4} \\
\Rightarrow \cos \gamma=\frac{1}{2} \Rightarrow \gamma=60^{\circ} \\
\end{array}
$$
$$
\begin{aligned}
\Rightarrow & \cos \gamma=\frac{1}{2} \Rightarrow \gamma=60^{\circ}
\end{aligned}
$$
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