If a line makes angle $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the $X$-axis and $Y$-axis respectively, then the angle made by the line with the $Z$-axis is

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Aicharya · Accepted Answer

We know that, $\begin{array}{rlrl} & & \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \ 	herefore & & \cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{4}+\cos ^2 \gamma=1 \ \Rightarrow & \left(\frac{1}{2}ight)^2+\left(\frac{1}{\sqrt{2}}ight)^2+\cos ^2 \gamma=1 \ \Rightarrow & \cos ^2 \gamma=1-\frac{1}{4}-\frac{1}{2}=\frac{1}{4} \ \Rightarrow & \cos \gamma=\frac{1}{2}=\cos \frac{\pi}{3} \ \Rightarrow & \gamma=\frac{\pi}{3}\end{array}$

If a line makes angle $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the $X$-axis and $Y$-axis respectively, then…

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