Let's break down the problem and solve it step by step.

Part (a) Finding the value of $\theta$

Given:

• Radius of the large circle $R = 8 \, \text{cm}$
• Radius of the smaller circle $r = 2 \, \text{cm}$
• Angle $\text{AOB} = 2\theta$ radians

The smaller circle is inscribed in the sector $AOB$, touching the lines $OA$ and $OB$, and the arc $AB$. The distance from the center $O$ to the center of the smaller circle is $R - r = 8 - 2 = 6 \, \text{cm}$.

For the inscribed circle, the distance from $O$ to the center of the smaller circle (let's call it $C$) can also be represented as the hypotenuse of a right triangle where:

• The distance from $O$ to the tangent point on $OA$ is $OC\cos(\theta)$,
• The distance from $C$ to the tangent point on $AB$ is $OC\sin(\theta)$.

Given $OC = R - r$, we have: $OC = 6 \, \text{cm}$

The radius $r$ can also be expressed in terms of $\theta$ using trigonometric relationships: $r = R(1 - \cos(\theta))$

However, by focusing on the geometry: $\frac{r}{OC} = \sin(\theta)$ $\sin(\theta) = \frac{r}{OC} = \frac{2}{6} = \frac{1}{3}$ $\theta = \arcsin\left(\frac{1}{3}\right)$

Thus, $2\theta$ is approximately: $\theta \approx \arcsin\left(\frac{1}{3}\right) \approx 0.3398 \, \text{radians}$

Part (b) Finding the area of the shaded region

The shaded region is the area of the sector $AOB$ minus the area of the inscribed circle.

• Area of sector $AOB$: $\text{Area of sector} = \frac{1}{2} R^2 \times 2\theta = R^2 \theta = 8^2 \times 0.3398 \approx 21.7472 \, \text{cm}^2$

• Area of the inscribed circle: $\text{Area of smaller circle} = \pi r^2 = \pi \times 2^2 = 4\pi \approx 12.5664 \, \text{cm}^2$

• Area of the shaded region: $\text{Shaded area} = \text{Area of sector} - \text{Area of inscribed circle} \approx 21.7472 - 12.5664 = 9.1808 \, \text{cm}^2$

To 3 significant figures, the area of the shaded region is: $\text{Shaded area} \approx 9.18 \, \text{cm}^2$

Summary:

• (a) The value of $\theta$ is approximately $0.3398$ radians (to 4 significant figures).
• (b) The area of the shaded region is approximately $9.18 \, \text{cm}^2$ (to 3 significant figures).

Would you like to see more details on any part of this solution?

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Here are some additional related questions for you:

1. How would the area of the shaded region change if the radius of the smaller circle were increased?
2. What would the area of the shaded region be if the angle $2\theta$ were doubled?
3. How would the value of $\theta$ change if the radius of the smaller circle were halved?
4. Can you derive a general formula for the area of the shaded region in terms of $R$, $r$, and $\theta$?
5. What is the relationship between the angle $\theta$ and the radius of the smaller circle?
6. How does the area of the sector change if $\theta$ is increased by $10\%$?
7. What is the perimeter of the shaded region?
8. Can you find the length of the arc $AB$ for the given value of $\theta$?

Tip: When dealing with problems involving sectors and circles, understanding trigonometric relationships is key to finding distances and areas effectively.