To find the angle subtended at the center of the circle by the chord of length $\frac{6}{3} \, \text{cm} = 2 \, \text{cm}$, we can use the relationship between the length of a chord, the radius of the circle, and the angle subtended at the center.

Step 1: Find the Radius of the Circle

Given that a chord of length $6 \, \text{cm}$ subtends an angle of $60^\circ$:

1. The formula for the length of a chord $c$ in terms of the radius $r$ and the angle $\theta$ in radians is:

   $c = 2r \sin\left(\frac{\theta}{2}\right)$

2. Convert $60^\circ$ to radians:

   $60^\circ = \frac{\pi}{3} \, \text{radians}$

3. Substitute the values into the formula:

   $6 = 2r \sin\left(\frac{\pi}{6}\right)$

   Knowing that $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$:

   $6 = 2r \cdot \frac{1}{2} \implies 6 = r$

So, the radius $r$ of the circle is $6 \, \text{cm}$.

Step 2: Find the Angle for the 2 cm Chord

Now, we will find the angle $\theta'$ subtended by the chord of length $2 \, \text{cm}$:

1. Use the chord length formula again:

   $2 = 2r \sin\left(\frac{\theta'}{2}\right)$

2. Substitute $r = 6$:

   $2 = 2 \cdot 6 \sin\left(\frac{\theta'}{2}\right) \implies 2 = 12 \sin\left(\frac{\theta'}{2}\right)$

3. Solve for $\sin\left(\frac{\theta'}{2}\right)$:

   $\sin\left(\frac{\theta'}{2}\right) = \frac{2}{12} = \frac{1}{6}$

4. Find $\frac{\theta'}{2}$:

   $\frac{\theta'}{2} = \arcsin\left(\frac{1}{6}\right)$

5. Therefore, $\theta'$:

   $\theta' = 2 \arcsin\left(\frac{1}{6}\right)$

Step 3: Calculate the Angle

Using a calculator:

$\theta' \approx 2 \cdot 0.167 \approx 0.334 \, \text{radians}$

To convert this back to degrees:

$\theta' \approx 0.334 \cdot \frac{180}{\pi} \approx 19.2^\circ$

Conclusion

The angle subtended by the chord of length $2 \, \text{cm}$ at the center of the circle is approximately $19.2^\circ$.

Would you like more details or do you have any questions? Here are some related questions you might consider:

1. How would the angle change if the chord length was increased to 4 cm?
2. What happens to the angle if the radius of the circle is doubled?
3. Can you derive the chord length for an angle of $45^\circ$?
4. How do you calculate the area of the sector formed by a chord?
5. What is the relationship between the radius and chord lengths for different angles?

Tip: Always convert angles to radians when using trigonometric functions for consistency.