A positive acute angle is divided into two parts whose tangents are `(1)/(8)and(7)/(9)` . What is the value of this angle ?

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(12)`

Explanation

To solve the problem, we need to find the acute angle \( \theta \) that is divided into two parts \( \phi \) and \( \psi \) such that \( \tan \phi = \frac{1}{8} \) and \( \tan \psi = \frac{7}{9} \).

### Step-by-Step Solution:

1. **Define the angles**:
Let \( \phi \) and \( \psi \) be the two parts of the angle \( \theta \). We know:
\[ \tan \phi = \frac{1}{8} \quad \text{and} \quad \tan \psi = \frac{7}{9} \]

2. **Use the tangent addition formula**:
The tangent of the sum of two angles is given by:
\[ \tan(\phi + \psi) = \frac{\tan \phi + \tan \psi}{1 - \tan \phi \tan \psi} \]
Therefore, we can write:
\[ \tan \theta = \tan(\phi + \psi) = \frac{\tan \phi + \tan \psi}{1 - \tan \phi \tan \psi} \]

3. **Substitute the values of \( \tan \phi \) and \( \tan \psi \)**:
Substitute \( \tan \phi = \frac{1}{8} \) and \( \tan \psi = \frac{7}{9} \):
\[ \tan \theta = \frac{\frac{1}{8} + \frac{7}{9}}{1 - \left(\frac{1}{8} \cdot \frac{7}{9}\right)} \]

4. **Calculate the numerator**:
To add \( \frac{1}{8} \) and \( \frac{7}{9} \), we need a common denominator. The least common multiple of 8 and 9 is 72:
\[ \frac{1}{8} = \frac{9}{72} \quad \text{and} \quad \frac{7}{9} = \frac{56}{72} \]
Therefore,
\[ \tan \phi + \tan \psi = \frac{9}{72} + \frac{56}{72} = \frac{65}{72} \]

5. **Calculate the denominator**:
Now calculate \( 1 - \tan \phi \tan \psi \):
\[ \tan \phi \tan \psi = \frac{1}{8} \cdot \frac{7}{9} = \frac{7}{72} \]
Thus,
\[ 1 - \tan \phi \tan \psi = 1 - \frac{7}{72} = \frac{72 - 7}{72} = \frac{65}{72} \]

6. **Combine the results**:
Now substitute back into the formula for \( \tan \theta \):
\[ \tan \theta = \frac{\frac{65}{72}}{\frac{65}{72}} = 1 \]

7. **Find the angle \( \theta \)**:
Since \( \tan \theta = 1 \), we know:
\[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \text{ radians} = 45^\circ \]

### Final Answer:
The value of the angle \( \theta \) is \( 45^\circ \).