Figure shows the sector, AOB of a circle with centre O and radius 8 cm.
A circle of radius 2cm touches the lines OA and OB and the arc AB.
Angle AOB is 2θ radians, 0 < θ <
π
4
.
(a) Find, to 4 significant figures, the value of θ

To solve this problem, we'll use geometry and trigonometry to relate the angle \( \theta \) to the dimensions given. The problem involves a sector of a circle with center \( O \) and radius 8 cm, and a smaller circle of radius 2 cm that touches both the lines \( OA \) and \( OB \) as well as the arc \( AB \).

### Step 1: Relationship Between the Radius and the Angle
Let the smaller circle touch \( OA \) at point \( P \), \( OB \) at point \( Q \), and the arc \( AB \) at point \( R \).

Since the circle touches \( OA \) and \( OB \), the distance from \( O \) to the center of the smaller circle \( C \) is \( 8 - 2 = 6 \) cm. Additionally, because the smaller circle is tangent to both \( OA \) and \( OB \), \( C \) lies on the angle bisector of \( \angle AOB \), which implies that \( \angle AOC = \theta \).

### Step 2: Using the Distance Formula
The distance from the center of the smaller circle \( C \) to line \( OA \) (or \( OB \)) is equal to the radius of the smaller circle, which is 2 cm. This distance can also be expressed as \( 6 \sin \theta \).

So, we have:
\[
6 \sin \theta = 2
\]
\[
\sin \theta = \frac{2}{6} = \frac{1}{3}
\]

### Step 3: Solving for \( \theta \)
To find \( \theta \), take the inverse sine:
\[
\theta = \arcsin \left( \frac{1}{3} \right)
\]

### Step 4: Calculate \( \theta \) to 4 Significant Figures
We can calculate \( \theta \) using a calculator or software:
\[
\theta \approx \arcsin \left( \frac{1}{3} \right) \approx 0.3398 \text{ radians}
\]

Thus, the value of \( \theta \) to 4 significant figures is \( \theta \approx 0.3398 \) radians.

Would you like to go over any details or have further questions on this? Here are some related questions you might find interesting:

1. How would the solution change if the radius of the smaller circle was different?
2. What is the area of the sector \( AOB \) given \( \theta = 0.3398 \) radians?
3. How can you derive the relationship between \( \theta \) and the radius of the smaller circle?
4. What is the length of the arc \( AB \) when \( \theta = 0.3398 \) radians?
5. What happens to the angle \( \theta \) if the radius of the smaller circle is halved?
6. Can this problem be solved using the cosine rule instead of the sine rule?
7. How would the placement of the smaller circle change if \( \theta \) were larger than \( \frac{\pi}{4} \)?
8. Can this problem be generalized to any radius for the sector's circle?

**Tip:** When working with trigonometric functions, it's often useful to sketch the problem to better understand the relationships between the angles and distances.

Learn how to calculate θ in a circle sector problem involving a tangent circle. This high school-level problem uses geometry and trigonometry to find θ, given specific circle dimensions and angle conditions.

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