The critical angle of a transparent crystal is `45^(@)`. Then its polarizing angle is

Options

A. `theta = tan^(-1)(sqrt(2))`
B. `theta = sin^(-1)(sqrt(2))`
C. `theta = cos^(-1)((1)/(sqrt(2)))`
D. `theta = cot^(-1)(sqrt(2))`

Answer

To find the polarizing angle (P) of a transparent crystal when the critical angle (C) is given as \(45^\circ\), we can use the relationship between the critical angle and the polarizing angle. The relationship is given by the formula:

\[ \tan P = \frac{1}{\sin C} \]

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The critical angle \(C\) is given as \(45^\circ\).

2. **Use the Relationship**:
- We will use the formula \( \tan P = \frac{1}{\sin C} \).

3. **Calculate \(\sin C\)**:
- Since \(C = 45^\circ\), we know that:
\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \]

4. **Substitute into the Formula**:
- Substitute \(\sin C\) into the equation:
\[ \tan P = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

5. **Find the Polarizing Angle \(P\)**:
- Now, we need to find \(P\) using the inverse tangent function:
\[ P = \tan^{-1}(\sqrt{2}) \]

6. **Conclusion**:
- Therefore, the polarizing angle \(P\) is:
\[ P = \tan^{-1}(\sqrt{2}) \]

### Final Answer:
The polarizing angle is \( \tan^{-1}(\sqrt{2}) \).

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