Question

Explain how you can estimate the endpoints of each of the following angles inside a (unit) circle. You may use the scratch space provided to show work, but please also explain inside the textbox.1. 2 radians 2. `\frac{\pi}{2}` radians 3. 3.25 radians

Answer 1

To estimate the endpoints of angles in a unit circle, we can use the concept of radians to determine the corresponding coordinates on the circle. Let's explain how to estimate the endpoints of each of the specified angles step-by-step.

### 1. 2 Radians

- **Understanding Radians:** One complete revolution (360 degrees) is equal to \(2\pi\) radians (approximately 6.28). Therefore, 2 radians is a little less than \( \frac{1}{3} \times 2\pi \) or about 114.6 degrees.
- **Finding Endpoint Coordinates:** The coordinates of an angle \(\theta\) in a unit circle can be computed using:
\[
(x, y) = (\cos(\theta), \sin(\theta))
\]
For \(\theta = 2\) radians:
\[
x = \cos(2) \approx -0.416 \\
y = \sin(2) \approx 0.909
\]
**Endpoint Estimate:** Thus, the endpoint for \(2\) radians on the unit circle is approximately \((-0.416, 0.909)\).

### 2. \(\frac{\pi}{2}\) Radians

- **Understanding Radians:** \(\frac{\pi}{2}\) radians is equivalent to \(90\) degrees, which points directly upwards on the unit circle.
- **Finding Endpoint Coordinates:** Using the formulas:
\[
x = \cos\left(\frac{\pi}{2}\right) = 0 \\
y = \sin\left(\frac{\pi}{2}\right) = 1
\]
**Endpoint Estimate:** Therefore, the endpoint for \(\frac{\pi}{2}\) radians on the unit circle is \((0, 1)\).

### 3. 3.25 Radians

- **Understanding Radians:** To find the equivalent in degrees, \(3.25\) radians is roughly:
\[
\frac{3.25 \cdot 180}{\pi} \approx 186.5 \text{ degrees}
\]
This angle lies in the third quadrant of the unit circle since it is greater than \(\pi\) (or 180 degrees) but less than \(3\pi/2\) (or 270 degrees).
- **Finding the Equivalent Angle:** The reference angle can be found by subtracting \(\pi\):
\[
3.25 - \pi \approx 0.57
\]
- **Calculating Coordinates:** Using this value in terms of functions:
\[
x = \cos(3.25) \approx -0.116 \\
y = \sin(3.25) \approx -0.995
\]
**Endpoint Estimate:** Therefore, the endpoint for \(3.25\) radians on the unit circle is approximately \((-0.116, -0.995)\).

### Summary of Endpoints:

- For \(2\) radians: Approx \((-0.416, 0.909)\)
- For \(\frac{\pi}{2}\) radians: Exact \((0, 1)\)
- For \(3.25\) radians: Approx \((-0.116, -0.995)\)