Question
A line makes the same angle $$\alpha $$ with each of the $$x$$ and $$y$$ axes. If the angle $$\theta $$, which it makes with the $$z$$-axis, is such that $${\sin ^2}\theta = 2\,{\sin ^2}\alpha ,$$ then what is the value of $$\alpha \,?$$
A.
$$\frac{\pi }{4}$$
B.
$$\frac{\pi }{6}$$
C.
$$\frac{\pi }{3}$$
D.
$$\frac{\pi }{2}$$
Answer :
$$\frac{\pi }{4}$$
Solution :
$$\eqalign{
& {\text{Since, }}{l^2} + {m^2} + {n^2} = 1 \cr
& \therefore \,{\cos ^2}\alpha + {\cos ^2}\alpha + {\cos ^2}\theta = 1......\left( {\text{i}} \right) \cr} $$
($$\because \,A$$ line makes the same angle $$\alpha $$ with $$x$$ and $$y$$-axes and $$\theta $$ with $$z$$-axis)
$$\eqalign{
& {\text{Also, }}{\sin ^2}\theta = 2\,{\sin ^2}\alpha \cr
& \Rightarrow 1 - {\cos ^2}\theta = 2\left( {1 - {{\cos }^2}\alpha } \right)\,\,\,\left( {\because {{\sin }^2}A + {{\sin }^2}A = 1} \right) \cr
& \Rightarrow {\cos ^2}\theta = 2\,{\cos ^2}\alpha - 1......\left( {{\text{ii}}} \right) \cr
& \therefore \,{\text{From equations }}\left( {\text{i}} \right)\,{\text{and }}\left( {{\text{ii}}} \right), \cr
& 2\,{\cos ^2}\alpha + 2{\cos ^2}\alpha - 1 = 1 \cr
& \Rightarrow 4\,{\cos ^2}\alpha = 2 \cr
& \Rightarrow \cos \,\alpha = \pm \frac{1}{{\sqrt 2 }} \cr
& \Rightarrow \alpha = \frac{\pi }{4},\,\frac{{3\pi }}{4} \cr} $$