Question

lf a line makes angles ${45}^o,{60}^o$ with oy, oz respectively where $O=(0,0,0)$ then the sine of the angle made by that line wlth ox is

• A. ${60}^o$
• B. $\frac{1}{2}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\sqrt{3}$

Answer 1

Answer: (C)

$\frac{\sqrt{3}}{2}$

Let the line makes an angle ${\theta }_1,{\theta }_2,{\theta }_3$ respectively with x,y and z axes.

Then direction cosines given by:

$\cos {\theta }_1=\alpha ,\cos {\theta }_2=\beta ,\cos {\theta }_3=\gamma$ and they satisfy the formula ${\alpha }^2+{\beta }^2+{\gamma }^2=1$.

Hence, by the given information we get,

${\cos }^2{\theta }_1=\frac{1}{4}$.

$\Rightarrow \cos {\theta }_1=\pm \frac{1}{2}$

$\Rightarrow {\theta }_1={60}^0$ or ${120}^0$.

In any case, $\sin {\theta }_1=\frac{\sqrt{3}}{2}$