Let p and q be the length of two chords of a circle which subtend angles `36^@ and 60^@` respectively at the centre of the circle . Then , the angle (in radian) subtended by the chord of length p + q at the centre of the circle is (use ` pi=3.1` )

Answer

To solve the problem, we need to find the angle subtended by the chord of length \( p + q \) at the center of the circle, given that the lengths of the chords \( p \) and \( q \) subtend angles of \( 36^\circ \) and \( 60^\circ \) respectively.

### Step-by-Step Solution:

1. **Convert Angles to Radians:**
- The angle \( 36^\circ \) in radians is given by:
\[ 36^\circ = \frac{36 \times \pi}{180} = \frac{36\pi}{180} = \frac{\pi}{5} \text{ radians} \]
- The angle \( 60^\circ \) in radians is given by:
\[ 60^\circ = \frac{60 \times \pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3} \text{ radians} \]

2. **Use the Chord Length Formula:**
- The length of a chord \( L \) that subtends an angle \( \theta \) at the center of a circle of radius \( r \) is given by:
\[ L = 2r \sin\left(\frac{\theta}{2}\right) \]
- For chord \( p \) subtending \( 36^\circ \):
\[ p = 2r \sin\left(\frac{36^\circ}{2}\right) = 2r \sin(18^\circ) \]
- For chord \( q \) subtending \( 60^\circ \):
\[ q = 2r \sin\left(\frac{60^\circ}{2}\right) = 2r \sin(30^\circ) = 2r \cdot \frac{1}{2} = r \]

3. **Express \( p + q \):**
- Now, we can express \( p + q \):
\[ p + q = 2r \sin(18^\circ) + r = r(2\sin(18^\circ) + 1) \]

4. **Find the Angle Subtended by Chord \( p + q \):**
- Let \( \theta \) be the angle subtended by the chord \( p + q \):
\[ p + q = 2r \sin\left(\frac{\theta}{2}\right) \]
- Setting the two expressions for \( p + q \) equal:
\[ r(2\sin(18^\circ) + 1) = 2r \sin\left(\frac{\theta}{2}\right) \]
- Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
\[ 2\sin(18^\circ) + 1 = 2\sin\left(\frac{\theta}{2}\right) \]
- Rearranging gives:
\[ \sin\left(\frac{\theta}{2}\right) = \sin(18^\circ) + \frac{1}{2} \]

5. **Calculate \( \theta \):**
- Using the known value \( \sin(18^\circ) \approx 0.309 \):
\[ \sin\left(\frac{\theta}{2}\right) \approx 0.309 + 0.5 = 0.809 \]
- Now, find \( \frac{\theta}{2} \):
\[ \frac{\theta}{2} = \sin^{-1}(0.809) \]
- Using a calculator, we find:
\[ \frac{\theta}{2} \approx 0.927 \text{ radians} \]
- Therefore:
\[ \theta \approx 2 \times 0.927 \approx 1.854 \text{ radians} \]

6. **Final Result:**
- The angle subtended by the chord of length \( p + q \) at the center of the circle is approximately \( \theta \approx 1.854 \text{ radians} \).