The least angle of deviation for a glass prism is equal to its refracting angle. The refractive index of glass is 1.5. Then the angle of the prism is

Options

A. `2cos^(-1)(3/4)`
B. `sin^(-1)(3/4)`
C. `sin^(-1)(3/2)`
D. `cos^(-1)(3/2)`

Answer

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

### Step-by-Step Solution:

1. **Understanding 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 least angle of deviation (D) using the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

2. **Substituting the Given Values**:
Since it is given that the least angle of deviation (D) is equal to the angle of the prism (A), we can substitute D = A into 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. **Using the Given Refractive Index**:
We know that the refractive index (μ) is given as 1.5. So we can write:
\[ 1.5 = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

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

5. **Using the Double Angle Formula**:
We can use the double angle formula for sine:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
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. **Dividing 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. **Solving for \(\cos\left(\frac{A}{2}\right)\)**:
This simplifies to:
\[ \cos\left(\frac{A}{2}\right) = \frac{1.5}{2} = 0.75 \]

8. **Finding \(\frac{A}{2}\)**:
Now, we can find \(\frac{A}{2}\) using the inverse cosine function:
\[ \frac{A}{2} = \cos^{-1}(0.75) \]

9. **Calculating A**:
Therefore, the angle of the prism (A) is:
\[ A = 2 \cos^{-1}(0.75) \]

### Final Answer:
The angle of the prism is \( A = 2 \cos^{-1}(0.75) \).

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