Given that `Delta = 6, r_(1) = 2,r_(2)=3, r_(3) = 6` Difference between the greatest and the least angles is

Options

A. `cos^(-1).(4)/(5)`
B. `tan^(-1).(3)/(4)`
C. `cos^(-1).(3)/(5)`
D. none of these

Explanation

To solve the problem, we need to find the difference between the greatest and the least angles of a triangle given the values of the area (Δ) and the inradii (r1, r2, r3).

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Area of the triangle (Δ) = 6
- Inradius values: r1 = 2, r2 = 3, r3 = 6

2. **Use the Formula for Inradius**:
The inradius \( r \) of a triangle can be expressed in terms of the area \( Δ \) and the semi-perimeter \( s \):
\[ r = \frac{Δ}{s} \]
For each inradius, we can write:
- \( r_1 = \frac{Δ}{s - a} \)
- \( r_2 = \frac{Δ}{s - b} \)
- \( r_3 = \frac{Δ}{s - c} \)

3. **Set Up the Equations**:
Plugging in the values, we have:
- \( 2 = \frac{6}{s - a} \) → \( s - a = 3 \) → \( s = a + 3 \) (Equation 1)
- \( 3 = \frac{6}{s - b} \) → \( s - b = 2 \) → \( s = b + 2 \) (Equation 2)
- \( 6 = \frac{6}{s - c} \) → \( s - c = 1 \) → \( s = c + 1 \) (Equation 3)

4. **Equate the Semi-Perimeter**:
From the three equations, we can equate the expressions for \( s \):
\[ a + 3 = b + 2 = c + 1 \]

5. **Express \( a, b, c \) in Terms of Each Other**:
Rearranging gives:
- From Equation 1: \( a = s - 3 \)
- From Equation 2: \( b = s - 2 \)
- From Equation 3: \( c = s - 1 \)

6. **Substitute and Solve**:
Using \( s = a + 3 \):
- Substitute \( b \) and \( c \):
- \( b = (a + 3) - 2 = a + 1 \)
- \( c = (a + 3) - 1 = a + 2 \)

7. **Find the Angles**:
Since we have \( a, b, c \) in terms of \( a \):
- \( a = 3 \)
- \( b = 4 \)
- \( c = 5 \)

8. **Determine the Greatest and Least Angles**:
In a triangle, the angle opposite the longest side is the greatest angle. Here, side lengths are:
- \( a = 3 \)
- \( b = 4 \)
- \( c = 5 \) (hypotenuse)

Thus, the greatest angle \( C \) is \( 90^\circ \) (as it is a right triangle), and the smallest angle \( A \) can be calculated using the sine rule or cosine rule.

9. **Calculate the Smallest Angle**:
Using the sine function:
\[ \sin A = \frac{a}{c} = \frac{3}{5} \]
Therefore, \( A = \sin^{-1}\left(\frac{3}{5}\right) \).

10. **Find the Difference**:
The difference between the greatest and the least angle is:
\[ 90^\circ - A = 90^\circ - \sin^{-1}\left(\frac{3}{5}\right) \]

### Final Answer:
The difference between the greatest and the least angles is:
\[ 90^\circ - \sin^{-1}\left(\frac{3}{5}\right) \]