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प्रश्न
If a line makes angle \[\frac{\pi}{3} \text{ and } \frac{\pi}{4}\] with x-axis and y-axis respectively, then the angle made by the line with z-axis is
पर्याय
π/2
π/3
π/4
5π/12
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उत्तर
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,
\[\alpha = \frac{\pi}{3}\]
\[\beta = \frac{\pi}{4}\]
Now,
\[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]
\[ \Rightarrow \cos^2 \frac{\pi}{3} + \cos^2 \frac{\pi}{4} + \cos^2 \gamma = 1\]
\[ \Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1\]
\[ \Rightarrow \cos^2 \gamma = 1 - \frac{3}{4}\]
\[ \Rightarrow \cos^2 \gamma = \frac{1}{4}\]
\[ \Rightarrow \cos \gamma = \frac{1}{2}\]
\[ \Rightarrow \gamma = \frac{\pi}{3}\]
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