Let `alpha, beta, gamma` be the angles made by a line with the positive directions of the axes of reference in three dimensions. If `theta` is the acute angle given by
`costheta=(cos^(2)alpha+cos^(2)beta+cos^(2)gamma)/(sin^(2)alpha+sin^(2)beta+sin^(2)gamma)`, then `theta` equal.

Options

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

Answer

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

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

### Step 1: Understand the Direction Cosines
The angles \( \alpha, \beta, \gamma \) are the angles made by a line with the positive directions of the x, y, and z axes, respectively. The direction cosines are defined as:
- \( \cos \alpha \)
- \( \cos \beta \)
- \( \cos \gamma \)

### Step 2: Use the Property of Direction Cosines
We know that the sum of the squares of the direction cosines is equal to 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 3: Express Sine in Terms of Cosine
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express the sine squares in terms of cosine squares:
\[ \sin^2 \alpha = 1 - \cos^2 \alpha \]
\[ \sin^2 \beta = 1 - \cos^2 \beta \]
\[ \sin^2 \gamma = 1 - \cos^2 \gamma \]

### Step 4: Substitute into the Denominator
Now substituting these into the denominator:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma) \]
This simplifies to:
\[ = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 3 - 1 = 2 \]

### Step 5: Substitute Back into the Cosine Equation
Now substituting back into the equation for \( \cos \theta \):
\[ \cos \theta = \frac{\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma}{\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma} = \frac{1}{2} \]

### Step 6: Find the Angle \( \theta \)
Now we need to find \( \theta \) such that:
\[ \cos \theta = \frac{1}{2} \]
The angle \( \theta \) that satisfies this equation in the range of acute angles is:
\[ \theta = 60^\circ \text{ or } \theta = \frac{\pi}{3} \text{ radians} \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = \frac{\pi}{3} \]