Step 1: Understanding the Concept:
In 3D space, the direction of a line is defined by the angles it makes with the positive directions of the x, y, and z axes. These angles are denoted as \(\alpha\), \(\beta\), and \(\gamma\). The cosines of these angles, \(l = \cos\alpha, m = \cos\beta, n = \cos\gamma\), are known as the direction cosines of the line.
Direction cosines are essentially the components of a unit vector pointing in the same direction as the line. Because they represent a unit vector, the sum of their squares must always equal \(1\). This geometric property is a fundamental identity in 3D coordinate geometry.
Key Formula or Approach:
The primary identity for direction cosines is:
\[ \cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1 \]
Given that the line makes equal angles with the axes, we have the condition \(\alpha = \beta = \gamma\).
Step 2: Detailed Explanation:
Substitute the equality condition into the fundamental identity:
\[ \cos^{2}\alpha + \cos^{2}\alpha + \cos^{2}\alpha = 1 \]
Combine the identical terms:
\[ 3\cos^{2}\alpha = 1 \]
Isolate the square of the cosine:
\[ \cos^{2}\alpha = \frac{1}{3} \]
Take the square root of both sides:
\[ \cos\alpha = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}} \]
The problem specifies that the angles are acute. For an acute angle (\(0^{\circ}<\alpha<90^{\circ}\)), the value of \(\cos\alpha\) must be positive. Therefore, we discard the negative root.
\[ \cos\alpha = \frac{1}{\sqrt{3}} \]
The absolute value of \(\frac{1}{\sqrt{3}}\) is simply \(\frac{1}{\sqrt{3}}\).
Step 3: Final Answer:
Applying the direction cosine identity \(l^2 + m^2 + n^2 = 1\) and setting all angles equal leads to \(3\cos^2\alpha = 1\). Solving for the acute angle cosine gives \(\cos\alpha = 1/\sqrt{3}\).