To sketch the angle \(\frac{2\pi}{3}\), we first recognize that this angle is in radians.
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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.
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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 \(\frac{\pi}{3}\).