Question

A line makes angles of measure 45° and 60° with the positive directions of the \( Y \) and \( Z \) axes respectively. Find the angle made by the line with the positive direction of the \( X \)-axis.

Hint:

To find the angle between a line and the coordinate axes, use the direction cosines and the identity \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \).

Answer 1

Step 1: Use the direction cosines formula.
Let the direction cosines of the line be \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \), where \( \alpha \), \( \beta \), and \( \gamma \) are the angles the line makes with the \( X \), \( Y \), and \( Z \) axes, respectively. From the given information: \[ \cos \beta = \cos 45^\circ = \frac{1}{\sqrt{2}}, \cos \gamma = \cos 60^\circ = \frac{1}{2} \]

Step 2: Apply the direction cosines equation.
The sum of the squares of the direction cosines is always 1: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Substitute the known values for \( \cos \beta \) and \( \cos \gamma \): \[ \cos^2 \alpha + \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 = 1 \] \[ \cos^2 \alpha + \frac{1}{2} + \frac{1}{4} = 1 \] \[ \cos^2 \alpha + \frac{3}{4} = 1 \] \[ \cos^2 \alpha = 1 - \frac{3}{4} = \frac{1}{4} \] \[ \cos \alpha = \frac{1}{2} \]

Step 3: Find the angle \( \alpha \).
Since \( \cos \alpha = \frac{1}{2} \), we find: \[ \alpha = \cos^{-1} \left( \frac{1}{2} \right) = 60^\circ \]

Final Answer: The angle made by the line with the positive direction of the \( X \)-axis is \( \boxed{60^\circ} \).