If `sintheta-costheta=(1)/(2)` the value of `sintheta+costheta` is

Options

A. `-2`
B. `+-2`
C. `(sqrt(7))/(2)`
D. `2`

Answer

To solve the problem, we need to find the value of \( \sin \theta + \cos \theta \) given that \( \sin \theta - \cos \theta = \frac{1}{2} \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ \sin \theta - \cos \theta = \frac{1}{2} \]

2. **Square both sides:**
\[ (\sin \theta - \cos \theta)^2 = \left(\frac{1}{2}\right)^2 \]
This simplifies to:
\[ \sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = \frac{1}{4} \]

3. **Use the Pythagorean identity:**
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, we can replace \( \sin^2 \theta + \cos^2 \theta \) in the equation:
\[ 1 - 2\sin \theta \cos \theta = \frac{1}{4} \]

4. **Rearrange the equation:**
\[ -2\sin \theta \cos \theta = \frac{1}{4} - 1 \]
This simplifies to:
\[ -2\sin \theta \cos \theta = \frac{1}{4} - \frac{4}{4} = -\frac{3}{4} \]

5. **Divide both sides by -2:**
\[ \sin \theta \cos \theta = \frac{3}{8} \]

6. **Now, we need to find \( \sin \theta + \cos \theta \). Use the identity:**
\[ (\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta \]
Substitute the known values:
\[ (\sin \theta + \cos \theta)^2 = 1 + 2\left(\frac{3}{8}\right) \]
This simplifies to:
\[ (\sin \theta + \cos \theta)^2 = 1 + \frac{6}{8} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \]

7. **Take the square root:**
\[ \sin \theta + \cos \theta = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2} \]

### Final Answer:
\[ \sin \theta + \cos \theta = \frac{\sqrt{7}}{2} \]