Angle of minimum deviation for a prism of refractive index 1.5 is equal to the angle of prism of given prism. Then, the angle is prism is….
`(sin 48^(@)36'=0.75)`

Options

A. `80^(@)`
B. `41^(@)24'`
C. `60^(@)`
D. `82^(@)48'`

Explanation

To solve the problem, we need to find the angle of the prism (A) given that the angle of minimum deviation (D) is equal to the angle of the prism and the refractive index (μ) is 1.5.

### Step-by-Step Solution:

1. **Understand the Relationship**:
We know that for a prism, the relationship between the refractive index (μ), the angle of the prism (A), and the angle of minimum deviation (D) is given by the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Given that \(D = A\), we can substitute \(D\) with \(A\) in the formula.

2. **Substitute D with A**:
Since \(D = A\), we can rewrite the equation as:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substitute the Value of μ**:
We know that the refractive index \(μ = 1.5\), so we substitute this into the equation:
\[ 1.5 = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

4. **Cross-Multiply**:
Rearranging gives us:
\[ \sin(A) = 1.5 \cdot \sin\left(\frac{A}{2}\right) \]

5. **Use the Double Angle Identity**:
We can use the double angle identity for sine, which states:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
Substituting this into our equation gives:
\[ 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) = 1.5 \cdot \sin\left(\frac{A}{2}\right) \]

6. **Divide by \(\sin\left(\frac{A}{2}\right)\)**:
Assuming \(\sin\left(\frac{A}{2}\right) \neq 0\), we can divide both sides by \(\sin\left(\frac{A}{2}\right)\):
\[ 2 \cos\left(\frac{A}{2}\right) = 1.5 \]

7. **Solve for \(\cos\left(\frac{A}{2}\right)\)**:
Dividing both sides by 2 gives:
\[ \cos\left(\frac{A}{2}\right) = \frac{1.5}{2} = 0.75 \]

8. **Find \(\frac{A}{2}\)**:
To find \(\frac{A}{2}\), we take the inverse cosine:
\[ \frac{A}{2} = \cos^{-1}(0.75) \]

9. **Calculate A**:
Using the known value \( \cos^{-1}(0.75) \), we can find:
\[ \frac{A}{2} = 48^\circ 36' \]
Therefore, multiplying by 2 gives:
\[ A = 2 \times 48^\circ 36' = 97^\circ 12' \]

### Final Answer:
The angle of the prism \(A\) is \(97^\circ 12'\).