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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

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Solution

## (b) π/3 If a line makes angles α, β and γ with the axes, then ${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1$. Here, $\alpha =\frac{\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}\beta =\frac{\mathrm{\pi }}{4}$ Now, ${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\frac{\mathrm{\pi }}{3}+{\mathrm{cos}}^{2}\frac{\mathrm{\pi }}{4}+{\mathrm{cos}}^{2}\gamma =1\phantom{\rule{0ex}{0ex}}⇒\frac{1}{4}+\frac{1}{2}+{\mathrm{cos}}^{2}\gamma =1\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\gamma =1-\frac{3}{4}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\gamma =\frac{1}{4}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}\gamma =\frac{1}{2}\phantom{\rule{0ex}{0ex}}⇒\gamma =\frac{\mathrm{\pi }}{3}$

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