A line lies in XZ-plane and makes an angle `60^(@)` with Z-axis, find its inclination with X-axis.

Explanation

Let `alpha, beta, gamma,` be the angles made by the line with the positiv directions of X-, Y- and Z-axes respectively.
Then `gamma = 60^(@)` .
since the lines in the XZ-plane, it is perpendicular to the Y-axis.
`therefore beta =90^(@)`
`thereforecos^(2)alpha+cos^(2)beta+cos^(2)gamma=1 `gives
`cos^(2)alpha+cos^(2)90^(@) + cos^(2)60^(@)=1`
`therefore cos^(2)alpha+(0)^(2)+((1)/(2))^(2)=1`
`therefore cos^(2)alpha = 1 - (1)/(4) = (3)/(4) " " therefore cos alpha = pm (sqrt(3))/(2)`
`therefore cos alpha=(sqrt(3))/(2) or cos alpha=-(sqrt(3))/(2)`
`therefore cos alpha = cos30^(@) or cosalpha = -cos 30^(@) = cos(180^(@)-30^(@)) = cos 150^(@)`
`therefore alpha = 30^(@) or 150^(@)`
`therefore` the inclination of the line with X-axis is `30^(@) or 150^(@)`.