The angle of minimum deviation for a glass prism with `mu=sqrt3` equals the refracting angle of the prism. What is the angle of the prism?

Explanation

To solve the problem, we need to find the angle of the prism (denoted as \( A \)) given that the angle of minimum deviation (\( D \)) is equal to the refracting angle of the prism, and the refractive index (\( \mu \)) of the prism is \( \sqrt{3} \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that the angle of minimum deviation \( D \) is equal to the angle of the prism \( A \).
- The refractive index \( \mu \) of the prism is given as \( \sqrt{3} \).

2. **Using the Formula for Refractive Index:**
- The formula relating the refractive index \( \mu \), the angle of the prism \( A \), and the angle of minimum deviation \( D \) is:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
- Since \( D = A \), we can substitute \( D \) in the formula:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substituting the Value of \( \mu \):**
- We substitute \( \mu = \sqrt{3} \):
\[ \sqrt{3} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

4. **Using the Trigonometric Identity:**
- We can express \( \sin(A) \) using the double angle formula:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
- Substituting this into our equation gives:
\[ \sqrt{3} = \frac{2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
- The \( \sin\left(\frac{A}{2}\right) \) terms cancel out (assuming \( \sin\left(\frac{A}{2}\right) \neq 0 \)):
\[ \sqrt{3} = 2 \cos\left(\frac{A}{2}\right) \]

5. **Solving for \( \cos\left(\frac{A}{2}\right) \):**
- Rearranging gives:
\[ \cos\left(\frac{A}{2}\right) = \frac{\sqrt{3}}{2} \]
- The angle whose cosine is \( \frac{\sqrt{3}}{2} \) is \( 30^\circ \):
\[ \frac{A}{2} = 30^\circ \]

6. **Finding the Angle of the Prism \( A \):**
- Multiplying both sides by 2:
\[ A = 60^\circ \]

### Final Answer:
The angle of the prism \( A \) is \( 60^\circ \).

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