Question
Explain how you can estimate the endpoints of each of the following angles inside a (unit) circle.
1. 2 radians
2. `\frac{\pi}{2}` radians
3. 3.25 radians
1. 2 radians
2. `\frac{\pi}{2}` radians
3. 3.25 radians
Answers
Answered by
GPT-4o mini
To estimate the endpoints of angles in a unit circle, we can visualize the circle in the Cartesian coordinate system. The unit circle has a radius of 1, and we will use the angle measurements in radians to find the corresponding points on the circle.
### 1. **2 radians**
- **Locate the angle:** To estimate 2 radians, we first note that it is a little more than \(\pi/2\) (which is approximately 1.57 radians) but less than \(\pi\) (which is approximately 3.14 radians). It's located in the second quadrant.
- **Find coordinates:** The angle wraps around the circle a bit, so:
\[
x = \cos(2) \quad \text{and} \quad y = \sin(2)
\]
- **Calculation:**
- Approximating these values gives:
- \(\cos(2) \approx -0.42\)
- \(\sin(2) \approx 0.91\)
- **Endpoint:** Thus, the endpoint for 2 radians is approximately \((-0.42, 0.91)\).
### 2. **\(\frac{\pi}{2}\) radians**
- **Locate the angle:** The angle \(\frac{\pi}{2}\) radians corresponds to 90 degrees, which is at the top of the unit circle.
- **Find coordinates:** The coordinates for \(\frac{\pi}{2}\) are:
\[
x = \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad y = \sin\left(\frac{\pi}{2}\right) = 1
\]
- **Endpoint:** So, the endpoint for \(\frac{\pi}{2}\) radians is \((0, 1)\).
### 3. **3.25 radians**
- **Locate the angle:** The angle 3.25 radians is slightly more than \(\pi\) (approximately 3.14 radians) and less than \(\frac{3\pi}{2}\) (approximately 4.71 radians). This places it in the third quadrant.
- **Find coordinates:** Now, to get the exact coordinates:
\[
x = \cos(3.25) \quad \text{and} \quad y = \sin(3.25)
\]
- **Calculation:**
- Approximating these values gives:
- \(\cos(3.25) \approx -0.11\)
- \(\sin(3.25) \approx -0.99\)
- **Endpoint:** Thus, the endpoint for 3.25 radians is approximately \((-0.11, -0.99)\).
In summary, the endpoints for the angles in the unit circle are as follows:
1. 2 radians: \((-0.42, 0.91)\)
2. \(\frac{\pi}{2}\) radians: \((0, 1)\)
3. 3.25 radians: \((-0.11, -0.99)\)
### 1. **2 radians**
- **Locate the angle:** To estimate 2 radians, we first note that it is a little more than \(\pi/2\) (which is approximately 1.57 radians) but less than \(\pi\) (which is approximately 3.14 radians). It's located in the second quadrant.
- **Find coordinates:** The angle wraps around the circle a bit, so:
\[
x = \cos(2) \quad \text{and} \quad y = \sin(2)
\]
- **Calculation:**
- Approximating these values gives:
- \(\cos(2) \approx -0.42\)
- \(\sin(2) \approx 0.91\)
- **Endpoint:** Thus, the endpoint for 2 radians is approximately \((-0.42, 0.91)\).
### 2. **\(\frac{\pi}{2}\) radians**
- **Locate the angle:** The angle \(\frac{\pi}{2}\) radians corresponds to 90 degrees, which is at the top of the unit circle.
- **Find coordinates:** The coordinates for \(\frac{\pi}{2}\) are:
\[
x = \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad y = \sin\left(\frac{\pi}{2}\right) = 1
\]
- **Endpoint:** So, the endpoint for \(\frac{\pi}{2}\) radians is \((0, 1)\).
### 3. **3.25 radians**
- **Locate the angle:** The angle 3.25 radians is slightly more than \(\pi\) (approximately 3.14 radians) and less than \(\frac{3\pi}{2}\) (approximately 4.71 radians). This places it in the third quadrant.
- **Find coordinates:** Now, to get the exact coordinates:
\[
x = \cos(3.25) \quad \text{and} \quad y = \sin(3.25)
\]
- **Calculation:**
- Approximating these values gives:
- \(\cos(3.25) \approx -0.11\)
- \(\sin(3.25) \approx -0.99\)
- **Endpoint:** Thus, the endpoint for 3.25 radians is approximately \((-0.11, -0.99)\).
In summary, the endpoints for the angles in the unit circle are as follows:
1. 2 radians: \((-0.42, 0.91)\)
2. \(\frac{\pi}{2}\) radians: \((0, 1)\)
3. 3.25 radians: \((-0.11, -0.99)\)