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^2theta=2sin^2alpha`, then what is the value of `alpha`?

Options

A. `pi//4`
B. `pi//6`
C. `pi//3`
D. `pi//2`

Answer

A

Explanation

Since `l^2+m^2+n^2=1`
`:. Cos^2alpha+cos^2alpha+cos^2theta=1 ……….(i)`
(`:'` A line makes the same angle `alpha` with x and y-axes and `theta` with z-axis)
Also, `sin^2theta=2sin^2alpha`
`rArr 1-cos^2theta=2(1-cos^2alpha)( :'sin^2A+cos^2A=1)`
`rArr cos^2theta=2cos^2alpha-1............(ii)`
`:." From Eq.(i) and (ii) "`
`2cos^2alpha+2cos^2alpha-1=1`
`rArr 4cos^2alpha=2rArr cos^2alpha=1/2`
`rArr cosalpha=pm1/sqrt2`
`rArr alpha=pi/4,(3pi)/4`