Question
Sketch the angle then find its reference angle
2pie over 3
2pie over 3
Answers
GPT-4o mini
To sketch the angle \(\frac{2\pi}{3}\), we first recognize that this angle is in radians.
1. **Sketching the angle**:
\(\frac{2\pi}{3}\) radians is equal to \(\frac{2}{3}\) of \(\pi\). Since \(\pi\) radians corresponds to 180 degrees, we can find \(\frac{2\pi}{3}\) in degrees:
\[
\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = \frac{2 \times 180}{3} = 120^\circ
\]
The angle \(120^\circ\) is located in the second quadrant of the unit circle.
2. **Finding the reference angle**:
The reference angle is found by considering how far the angle is from the x-axis, specifically from the nearest x-axis. In the second quadrant, the reference angle \( \theta_{\text{ref}} \) can be calculated as:
\[
\theta_{\text{ref}} = \pi - \frac{2\pi}{3}
\]
We can calculate this:
\[
\theta_{\text{ref}} = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{1\pi}{3} = \frac{\pi}{3}
\]
So, the reference angle for \(\frac{2\pi}{3}\) is \(\frac{\pi}{3}\).
1. **Sketching the angle**:
\(\frac{2\pi}{3}\) radians is equal to \(\frac{2}{3}\) of \(\pi\). Since \(\pi\) radians corresponds to 180 degrees, we can find \(\frac{2\pi}{3}\) in degrees:
\[
\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = \frac{2 \times 180}{3} = 120^\circ
\]
The angle \(120^\circ\) is located in the second quadrant of the unit circle.
2. **Finding the reference angle**:
The reference angle is found by considering how far the angle is from the x-axis, specifically from the nearest x-axis. In the second quadrant, the reference angle \( \theta_{\text{ref}} \) can be calculated as:
\[
\theta_{\text{ref}} = \pi - \frac{2\pi}{3}
\]
We can calculate this:
\[
\theta_{\text{ref}} = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{1\pi}{3} = \frac{\pi}{3}
\]
So, the reference angle for \(\frac{2\pi}{3}\) is \(\frac{\pi}{3}\).