If in two circle arcs of the same length subtend angle of `60^(@)` and `75^(@)` at the center , find the ratio of their radii.

Answer

To solve the problem, we need to find the ratio of the radii of two circles given that the lengths of their arcs are equal and they subtend angles of 60 degrees and 75 degrees at the center, respectively.

### Step-by-Step Solution:

1. **Convert Angles from Degrees to Radians:**
- The formula to convert degrees to radians is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
- For \(60^\circ\):
\[ 60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians} \]
- For \(75^\circ\):
\[ 75^\circ = 75 \times \frac{\pi}{180} = \frac{75\pi}{180} = \frac{5\pi}{12} \text{ radians} \]

2. **Use the Arc Length Formula:**
- The length of an arc \(L\) is given by:
\[ L = r \theta \]
- Let \(L_1\) be the arc length for the first circle and \(L_2\) for the second circle. Since the lengths are equal:
\[ L_1 = L_2 \]
- Therefore:
\[ R_1 \cdot \frac{\pi}{3} = R_2 \cdot \frac{5\pi}{12} \]

3. **Set Up the Equation:**
- From the above equation, we can simplify:
\[ R_1 \cdot \frac{\pi}{3} = R_2 \cdot \frac{5\pi}{12} \]
- Dividing both sides by \(\pi\) (assuming \(\pi \neq 0\)):
\[ R_1 \cdot \frac{1}{3} = R_2 \cdot \frac{5}{12} \]

4. **Rearranging the Equation:**
- Rearranging gives:
\[ \frac{R_1}{R_2} = \frac{5}{12} \cdot 3 \]
- This simplifies to:
\[ \frac{R_1}{R_2} = \frac{15}{12} \]

5. **Simplifying the Ratio:**
- Simplifying \(\frac{15}{12}\) gives:
\[ \frac{R_1}{R_2} = \frac{5}{4} \]

6. **Final Result:**
- Therefore, the ratio of the radii \(R_1\) to \(R_2\) is:
\[ R_1 : R_2 = 5 : 4 \]