# If a line $\overline {AB}$ makes angle $45^{\circ}$ with positive $x-axis$ and $120^{\circ}$ with positive $y-axis$, then the angle $\theta$ made by the line with positive $z-axis$ is ?

$(a)\;30^{0}\qquad(b)\;45^{0}\qquad(c)\;60^{0}\qquad(d)\;75^{0}$

Toolbox:
• If a line makes angles $\theta_1,\theta_2,\theta_3$ respectively with positive coordinate axes, then $cos^2 \theta_1+cos^2 \theta_2+cos^2 \theta_3=1$
Since it is given that $\overline {AB}$ makes $45^{\circ},\:120^{\circ}\:and\:\:\theta$ with the coordinate axes,
$cos^2 45^{\circ}+cos^2 120^{\circ}+cos^2 \theta=1$
$\Rightarrow\:\large\frac{1}{2}+\frac{1}{4}+$$cos^2\theta=1$
$\Rightarrow\:cos\theta=\large\frac{1}{2}$
$\therefore\:\theta=60^{\circ}$