Visible light of wavelenght `8000 xx 10^(-8)` cm falls normally on a single slit and produces a diffraction pattern. It is found that the Third diffraction minimum is at `45^(@)` from the central maximum. If the first minimum is produced at `theta_(1)` then `theta_(1) ` is close to :

Options

A. `40^(@)`
B. `35^(@)`
C. `30^(@)`
D. `15^(@)`

Explanation

To solve the problem, we need to find the angle \( \theta_1 \) at which the first diffraction minimum occurs, given that the third diffraction minimum is at \( 45^\circ \).

### Step-by-Step Solution:

1. **Understanding the Diffraction Condition**:
The condition for the minima in a single-slit diffraction pattern is given by the formula:
\[ n \lambda = d \sin \theta \]
where \( n \) is the order of the minimum, \( \lambda \) is the wavelength of the light, \( d \) is the width of the slit, and \( \theta \) is the angle at which the minimum occurs.

2. **Given Values**:
- Wavelength \( \lambda = 8000 \times 10^{-8} \) cm.
- The angle for the third minimum \( \theta_3 = 45^\circ \).
- For the third minimum, \( n = 3 \).

3. **Finding the Slit Width \( d \)**:
Using the formula for the third minimum:
\[ 3 \lambda = d \sin(45^\circ) \]
Since \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \), we can rewrite the equation as:
\[ d = \frac{3 \lambda}{\sin(45^\circ)} = \frac{3 \lambda}{\frac{\sqrt{2}}{2}} = 3 \sqrt{2} \lambda \]

4. **Substituting the Wavelength**:
Now substituting the value of \( \lambda \):
\[ d = 3 \sqrt{2} \times 8000 \times 10^{-8} \text{ cm} \]

5. **Finding the First Minimum**:
For the first minimum (\( n = 1 \)), we use the same formula:
\[ \lambda = d \sin(\theta_1) \]
Substituting the value of \( d \):
\[ \lambda = (3 \sqrt{2} \lambda) \sin(\theta_1) \]
Dividing both sides by \( \lambda \) (assuming \( \lambda \neq 0 \)):
\[ 1 = 3 \sqrt{2} \sin(\theta_1) \]
Rearranging gives:
\[ \sin(\theta_1) = \frac{1}{3 \sqrt{2}} \]

6. **Calculating \( \theta_1 \)**:
Now we find \( \theta_1 \):
\[ \theta_1 = \sin^{-1}\left(\frac{1}{3 \sqrt{2}}\right) \]

7. **Approximating \( \theta_1 \)**:
Using a calculator or trigonometric tables, we can find:
\[ \theta_1 \approx 15^\circ \]

### Final Answer:
Thus, \( \theta_1 \) is approximately \( 15^\circ \).