In three-dimensional space,a line $AB$ makes angles of $45^\circ$ and $120^\circ$ with the positive $x$-axis and $y$-axis,respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis,then $\theta = \dots^\circ$.

Let the direction cosines of two lines satisfy the equations $3l+2m+n=0$ and $2mn-3nl+5lm=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of the $x$-axis and $y$-axis,then the angle that the line makes with the positive direction of the $z$-axis is

If a line makes angles of $30^o$ and $45^o$ with $X-$ axis and $Y-$ axis,then the angle made by it with $Z-$ axis is

If the angle between the lines whose direction cosines are $\left(-\frac{2}{\sqrt{21}}, \frac{C}{\sqrt{21}}, \frac{1}{\sqrt{21}}\right)$ and $\left(\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}}, -\frac{6}{\sqrt{54}}\right)$ is $\frac{\pi}{2}$, then the value of $C$ is

If the angles made by a straight line with the coordinate axes are $\alpha, \frac{\pi}{2}-\alpha, \beta$, then $\beta$ is equal to