Angle of minimum deviation for a prism of refactive index 1.5, is equal to the angle of the prism. Then the angle of the prism is
`(cos 41^(@) = 0.75)`

Options

A. `62^(@)`
B. `41^(@)`
C. `82^(@)`
D. `31^(@)`

Answer

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 refractive index (μ) can be expressed in terms of the angle of the prism (A) and the angle of minimum deviation (D) using the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Since D = A, we can substitute D with A in the formula.

2. **Substitute D with A**: The equation now becomes:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

3. **Insert the value of μ**: Given that the refractive index (μ) is 1.5, we can write:
\[ 1.5 = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

4. **Express sin(A)**: We can express sin(A) in terms of sin(A/2):
\[ \sin(A) = 1.5 \cdot \sin\left(\frac{A}{2}\right) \]

5. **Use the double angle identity**: We know from trigonometric identities that:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
Therefore, we can equate:
\[ 1.5 \cdot \sin\left(\frac{A}{2}\right) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]

6. **Cancel sin(A/2)**: Assuming sin(A/2) is not zero, we can divide both sides by sin(A/2):
\[ 1.5 = 2 \cos\left(\frac{A}{2}\right) \]

7. **Solve for cos(A/2)**: Rearranging gives:
\[ \cos\left(\frac{A}{2}\right) = \frac{1.5}{2} = 0.75 \]

8. **Find A/2**: We know from the problem statement that:
\[ \cos(41^\circ) = 0.75 \]
Therefore, we can conclude:
\[ \frac{A}{2} = 41^\circ \]

9. **Calculate A**: Finally, multiplying both sides by 2 gives:
\[ A = 82^\circ \]

### Final Answer:
The angle of the prism (A) is \( 82^\circ \).