If `theta , phi` are acute, `sin theta = 1//2 , cos phi = 1//3`, then `[ theta + phi] in `

Options

A. `( pi //3 , pi //2)`
B. `( pi //2, 2pi // 3)`
C. `( 2pi //3, 5pi //6)`
D. `( 5pi //6, pi )`

Answer

To solve the problem, we need to find the range of the sum \( \theta + \phi \) given that \( \sin \theta = \frac{1}{2} \) and \( \cos \phi = \frac{1}{3} \), where \( \theta \) and \( \phi \) are acute angles.

### Step-by-step Solution:

1. **Find \( \theta \)**:
- We know that \( \sin \theta = \frac{1}{2} \).
- The angle \( \theta \) that satisfies this in the acute range is:
\[ \theta = \frac{\pi}{6} \]

2. **Find \( \phi \)**:
- We know that \( \cos \phi = \frac{1}{3} \).
- To find \( \phi \), we can use the inverse cosine function:
\[ \phi = \cos^{-1}\left(\frac{1}{3}\right) \]
- Since \( \phi \) is acute, we can find the approximate value of \( \phi \) using a calculator, but we will keep it in terms of \( \cos^{-1} \).

3. **Calculate \( \theta + \phi \)**:
- Now we can add \( \theta \) and \( \phi \):
\[ \theta + \phi = \frac{\pi}{6} + \cos^{-1}\left(\frac{1}{3}\right) \]

4. **Determine the range of \( \theta + \phi \)**:
- We know that \( \phi \) is between \( 0 \) and \( \frac{\pi}{2} \) since it is acute.
- The maximum value of \( \phi \) can be found by considering the maximum value of \( \cos^{-1}\left(\frac{1}{3}\right) \):
- Since \( \cos \phi \) is decreasing in the interval \( [0, \frac{\pi}{2}] \), \( \phi \) will be less than \( \frac{\pi}{2} \).
- Therefore, the maximum value of \( \theta + \phi \) can be calculated as:
\[ \theta + \phi < \frac{\pi}{6} + \frac{\pi}{2} = \frac{\pi}{6} + \frac{3\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

5. **Final Range**:
- Since \( \theta \) is fixed at \( \frac{\pi}{6} \) and \( \phi \) is positive and less than \( \frac{\pi}{2} \), we can conclude that:
\[ \frac{\pi}{6} < \theta + \phi < \frac{2\pi}{3} \]
- Thus, the final answer is:
\[ \theta + \phi \in \left(\frac{\pi}{6}, \frac{2\pi}{3}\right) \]