#### Question

A line makes an angle of 60° with each of *X*-axis and *Y*-axis. Find the acute angle made by the line with *Z*-axis.

#### Solution

\[ \text { It is given that a line makes an angle of 60° with both x - axis and y - axis } . \]

\[ \text{ Suppose the line makes an angle of } \alpha \text{ with the z - axis }. \]

\[ \Rightarrow l = cos\ 60° = \frac{1}{2}\]

\[m = \cos 60° = \frac{1}{2} \]

\[n = \cos \alpha\]

\[\text{ We know } \]

\[ l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^2 + \left( \cos \alpha \right)^2 = 1\]

\[ \Rightarrow \frac{1}{4} + \frac{1}{4} + \cos {}^2 \alpha = 1\]

\[ \Rightarrow \cos \alpha = \frac{1}{\sqrt{2}}\]

\[ \Rightarrow \alpha = 45°\]

\[ \text{ Thus, the line makes an angle of } 45° \text{ with the z - axis }. \]