The general value (s) of `theta` for which `cos 2 theta , 1/2` and `sin 2 theta` are A. P. is/are

Options

A. `n pi,n pi + pi/4`
B. `n pi, n pi + pi/2`
C. `n pi + pi/4`
D. `n pi`

Answer

To solve the problem, we need to find the general values of \( \theta \) for which \( \cos(2\theta), \frac{1}{2}, \) and \( \sin(2\theta) \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
For three terms \( A, B, C \) to be in A.P., the condition is:
\[ 2B = A + C \]
Here, \( A = \cos(2\theta) \), \( B = \frac{1}{2} \), and \( C = \sin(2\theta) \).

2. **Setting Up the Equation**:
Applying the A.P. condition:
\[ 2 \cdot \frac{1}{2} = \cos(2\theta) + \sin(2\theta) \]
Simplifying this gives:
\[ 1 = \cos(2\theta) + \sin(2\theta) \]

3. **Rearranging the Equation**:
Rearranging the equation:
\[ \cos(2\theta) + \sin(2\theta) = 1 \]

4. **Using Trigonometric Identities**:
We can use the identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \) and \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \):
\[ (1 - 2\sin^2(\theta)) + 2\sin(\theta)\cos(\theta) = 1 \]

5. **Simplifying the Equation**:
This simplifies to:
\[ -2\sin^2(\theta) + 2\sin(\theta)\cos(\theta) = 0 \]
Dividing through by 2:
\[ -\sin^2(\theta) + \sin(\theta)\cos(\theta) = 0 \]

6. **Factoring the Equation**:
Factoring out \( \sin(\theta) \):
\[ \sin(\theta)(\cos(\theta) - \sin(\theta)) = 0 \]

7. **Finding Solutions**:
This gives us two cases to solve:
- Case 1: \( \sin(\theta) = 0 \)
- Case 2: \( \cos(\theta) - \sin(\theta) = 0 \)

8. **Solving Case 1**:
For \( \sin(\theta) = 0 \):
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]

9. **Solving Case 2**:
For \( \cos(\theta) = \sin(\theta) \):
\[ \tan(\theta) = 1 \implies \theta = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]

10. **Final General Solutions**:
Combining both cases, the general solutions for \( \theta \) are:
\[ \theta = n\pi \quad \text{and} \quad \theta = \frac{\pi}{4} + n\pi \]

### Summary of Solutions:
The general values of \( \theta \) for which \( \cos(2\theta), \frac{1}{2}, \sin(2\theta) \) are in A.P. are:
\[ \theta = n\pi \quad \text{and} \quad \theta = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]