To solve for the angle $\theta$ in the given triangle, we can use the properties of the triangle and the given dimensions.

Step 1: Analyze the Triangle

The triangle appears to be a right triangle, where:

• The hypotenuse is $3$.
• One side is $2\sqrt{3}$.
• The other side is $1$.

Step 2: Apply Trigonometric Relationships

In a right triangle, the sine, cosine, and tangent functions relate the angle $\theta$ to the sides of the triangle.

1. Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
2. Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
3. Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step 3: Calculate $\theta$

Given the sides of the triangle:

• The side opposite to $\theta$ is $1$.
• The side adjacent to $\theta$ is $2\sqrt{3}$.
• The hypotenuse is $3$.

We can use the sine function: $\sin(\theta) = \frac{1}{3}$ To find $\theta$, take the inverse sine (arcsine) of both sides: $\theta = \arcsin\left(\frac{1}{3}\right)$

This will give us the value of $\theta$.

Step 4: Calculation

Let's compute the value: $\theta \approx \arcsin\left(\frac{1}{3}\right) \approx 19.47^\circ$

So, $\theta$ is approximately $19.47^\circ$.

Would you like further details or have any questions about the process?

Further Questions:

1. How would you find the other angles of the triangle?
2. What if the circle's radius $r$ was given? How would that change the problem?
3. Can you calculate $\theta$ using cosine or tangent instead of sine?
4. How does the radius of the inscribed circle relate to the sides of the triangle?
5. What is the area of this triangle?

Tip:

Always double-check the type of triangle (right, isosceles, etc.) before selecting a trigonometric function to find an angle.