Q88 / 160 · AP EAMCET 2023 (18 May Shift 2) · Mathematics
If a line $\mathrm{L}$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $\mathrm{X}$-axis and positive $\mathrm{Y}$-axis respectively, then the angle made by $\mathrm{L}$ with the positive direction of $\mathrm{Z}$-axis is
- A$\frac{\pi}{2}$
- Correct answerB$\frac{\pi}{3}$
- C$\frac{\pi}{4}$
- D$\frac{5 \pi}{12}$
Solution
Given $\Rightarrow \alpha=\frac{\pi}{3}, \beta=\frac{\pi}{4}$ Also, we have
$$
\begin{aligned}
& \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\
& \Rightarrow \cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{4}+\cos ^2 \gamma=1 \\
& \Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^2 \gamma=1 \\
& \Rightarrow \cos \gamma= \pm \frac{1}{2}
\end{aligned}
$$
For positive direction of $z$-axis
$$
\cos \gamma=\frac{1}{2} \Rightarrow \gamma=\frac{\pi}{3}
$$
Hence, the correct answer is option: 'B'.