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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.
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Solution
\[ \text { It is given that a line makes an angle of 60° with both x - axis and y - axis } . \]
\[ \text{ Suppose the line makes an angle of } \alpha \text{ with the z - axis }. \]
\[ \Rightarrow l = cos\ 60° = \frac{1}{2}\]
\[m = \cos 60° = \frac{1}{2} \]
\[n = \cos \alpha\]
\[\text{ We know } \]
\[ l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^2 + \left( \cos \alpha \right)^2 = 1\]
\[ \Rightarrow \frac{1}{4} + \frac{1}{4} + \cos {}^2 \alpha = 1\]
\[ \Rightarrow \cos \alpha = \frac{1}{\sqrt{2}}\]
\[ \Rightarrow \alpha = 45°\]
\[ \text{ Thus, the line makes an angle of } 45° \text{ with the z - axis }. \]
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