A diver at a depth of 12 m in water `(mu=4 //3)` sees the sky in a cone of semi-vertical angle

Options

A. `sin^(-1) (4//3)`
B. `tan^(-1) (4//3)`
C. `sin^(-1) (3//4)`
D. `90^(@)`

Answer

To solve the problem of finding the semi-vertical angle of the cone in which a diver at a depth of 12 m in water can see the sky, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A diver is located at a depth of 12 m in water. The refractive index of water is given as \( \mu = \frac{4}{3} \).
- The diver sees the sky in a cone of semi-vertical angle, which is determined by the critical angle for total internal reflection.

2. **Identifying the Critical Angle**:
- The critical angle (\( \theta_c \)) can be calculated using the formula:
\[ \theta_c = \sin^{-1}\left(\frac{1}{\mu}\right) \]
- Substituting the value of \( \mu \):
\[ \theta_c = \sin^{-1}\left(\frac{1}{\frac{4}{3}}\right) = \sin^{-1}\left(\frac{3}{4}\right) \]

3. **Calculating the Critical Angle**:
- Now, we need to find \( \sin^{-1}\left(\frac{3}{4}\right) \). This value can be computed using a calculator or trigonometric tables:
\[ \theta_c \approx 48.59^\circ \]

4. **Understanding the Semi-Vertical Angle**:
- The semi-vertical angle of the cone in which the diver can see the sky is equal to the critical angle \( \theta_c \).

5. **Conclusion**:
- Therefore, the semi-vertical angle of the cone is:
\[ \theta = \sin^{-1}\left(\frac{3}{4}\right) \approx 48.59^\circ \]

### Final Answer:
The semi-vertical angle of the cone in which the diver sees the sky is approximately \( 48.59^\circ \).

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