If `theta ` is an angle given by `cos theta = (cos ^(2) alpha + cos ^(2) beta + cos ^(2) gamma)/( sin ^(2) alpha + sin ^(2) beta + sin ^(2) gamma)` where `alpha, beta, gamma` are the equal angles made by line with the positive directions of the axes, then the measure of `theta` is

Options

A. `pi/3`
B. `pi/6`
C. `pi/2`
D. `pi/4`

Answer

To solve the problem, we need to find the measure of the angle \( \theta \) given the equation:

\[ \cos \theta = \frac{\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma}{\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma} \]

where \( \alpha, \beta, \gamma \) are equal angles made by a line with the positive directions of the axes.

### Step-by-step Solution:

1. **Understanding the Angles**:
Since \( \alpha, \beta, \gamma \) are equal angles, we can denote them as \( \alpha = \beta = \gamma = x \).

2. **Using Trigonometric Identities**:
We know that for any angle \( x \):
\[ \cos^2 x + \sin^2 x = 1 \]
Therefore, we can express \( \sin^2 x \) as:
\[ \sin^2 x = 1 - \cos^2 x \]

3. **Substituting the Equal Angles**:
Now substituting \( \alpha, \beta, \gamma \) with \( x \):
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 3\cos^2 x \]
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 3\sin^2 x \]

4. **Substituting into the Equation**:
Now substituting these into the equation for \( \cos \theta \):
\[ \cos \theta = \frac{3\cos^2 x}{3\sin^2 x} = \frac{\cos^2 x}{\sin^2 x} = \cot^2 x \]

5. **Using the Identity**:
We know that:
\[ \cot^2 x = \frac{1}{\tan^2 x} \]
Thus,
\[ \cos \theta = \cot^2 x \]

6. **Finding the Value of \( \theta \)**:
To find \( \theta \), we can use the fact that \( \cos \theta = 1 \) when \( x = 0 \) (which is not useful here). Instead, we need to find a specific angle where this holds true.

7. **Finding Specific Angles**:
We know that \( \cos \theta = \frac{1}{2} \) corresponds to \( \theta = \frac{\pi}{3} \) (or \( 60^\circ \)). Thus, we can conclude:
\[ \theta = \frac{\pi}{3} \]

### Final Answer:
The measure of \( \theta \) is \( \frac{\pi}{3} \).

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