Find coordinates on a circle from arc length and central angles

Expert intro: This problem mixes circular motion ideas with coordinate geometry. The key is to treat the arc length as the product of radius and central angle, then use standard circle coordinates from the angle at the center.

Detailed walkthrough

We are given:

• $O=(0,0)$
• $A=(3.6,0)$, so the radius is $r=3.6$
• $B=(2.99,2)$
• arc length $\overset{\frown}{BC}=14.436$

We want $\theta_1$, $\theta_2$, and point $C$.

a) Find $\theta_1$

Since $\angle AOB$ is the angle from the positive $x$-axis to $OB$,

$$\theta_1=\tan^{-1}\left(\frac{2}{2.99}\right)$$

So

$$\theta_1\approx 0.590\text{ rad}$$

b) Find $\theta_2$

Arc length formula:

$$s=r\theta$$

So

$$\theta_2=\frac{s}{r}=\frac{14.436}{3.6}=4.01\text{ rad}$$

c) Find coordinates of $C$

The angle to point $C$ is the total central angle:

$$\theta=\theta_1+\theta_2=\tan^{-1}\left(\frac{2}{2.99}\right)+\frac{14.436}{3.6}$$

Then

$$C=(3.6\cos\theta,\,3.6\sin\theta)$$

Using the approximate angle,

$$\theta\approx 0.590+4.01=4.60\text{ rad}$$

So

$$x\approx 3.6\cos(4.60)\approx -0.37$$ $$y\approx 3.6\sin(4.60)\approx -3.58$$

Answer

• $\theta_1\approx \boxed{0.590\text{ rad}}$
• $\theta_2=\boxed{4.01\text{ rad}}$
• $C\approx \boxed{(-0.37,-3.58)}$

💡 Pitfall guide

A common slip is forgetting that arc length uses radians, not degrees. Another one is missing the fact that the angle for point $C$ is the sum of the two central angles, not just the arc-based angle alone.

🔄 Real-world variant

If the arc from $B$ to $C$ were measured in the opposite direction, the angle for $C$ would be found by subtracting instead of adding. On a circle, direction matters, especially when the total angle wraps past $\pi$ or $2\pi$.

🔍 Related terms

arc length, central angle, unit circle