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If a line makes $45^{\circ},60^{\circ}$ with positive direction of axes $x$ and $y$ then the angle it makes with the $z$ axis is

\[\begin{array}{1 1}(1)30^{\circ}&(2)90^{\circ}\\(3)45^{\circ}&(4)60^{\circ}\end{array}\]

1 Answer

If $\alpha,\beta, \gamma$ are the angles made by a line with the positive direction of x-axis,y-axis ad z-axis the $\cos \alpha, \cos \beta, \cos \gamma$ are called the direction cosines of the line.
Also $ \cos ^2 \alpha + \cos ^2 \beta+ \cos ^2 \gamma=1$
Given $\alpha =45^{\circ}$
$\beta= 60^{\circ}$
$ \cos ^2 45^{\circ} + \cos ^2 60^{\circ}+ \cos ^2 \gamma=1$
$\bigg(\large\frac{1}{\sqrt {2}}\bigg)^2 +\bigg(\large\frac{1}{2}\bigg)^2$$+ \cos ^2 \gamma=1$
$\large\frac{1}{2}+\frac{1}{4} $$+\cos ^2 \gamma=1$
$\cos^2 \gamma=1- \large\frac{3}{4}=\frac{1}{4}$
$\cos \gamma=\large\frac{1}{2}$
Hence 4 is the correct answer.
answered May 12, 2014 by meena.p

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