Question

If a line makes angle $\frac{\mathrm{\pi }}{3}\mathrm{and}\frac{\mathrm{\pi }}{4}$ with x-axis and y-axis respectively, then the angle made by the line with z-axis is
(a) π/2
(b) π/3
(c) π/4
(d) 5π/12

Answer 1

(b) π/3

If a line makes angles α, β and γ with the axes, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$.

Here,

$$\begin{gathered} \alpha =\frac{\mathrm{\pi }}{3} \\ \beta =\frac{\mathrm{\pi }}{4} \end{gathered}$$

Now,

$$\begin{gathered} {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1 \\ \Rightarrow {\cos }^2\frac{\mathrm{\pi }}{3}+{\cos }^2\frac{\mathrm{\pi }}{4}+{\cos }^2\gamma =1 \\ \Rightarrow \frac{1}{4}+\frac{1}{2}+{\cos }^2\gamma =1 \\ \Rightarrow {\cos }^2\gamma =1-\frac{3}{4} \\ \Rightarrow {\cos }^2\gamma =\frac{1}{4} \\ \Rightarrow \cos \gamma =\frac{1}{2} \\ \Rightarrow \gamma =\frac{\mathrm{\pi }}{3} \end{gathered}$$