Identify the Quadrant of an Angle in Radians

Given an angle of $ \frac{4\pi}{3} $ radians, determine the quadrant in which the terminal side of the angle lies.

First, recall that the unit circle is divided into four quadrants:

1. Quadrant I:

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< \theta < \frac{\pi}{2}$

2. Quadrant II: $\frac{\pi}{2} < \theta < \pi$

3. Quadrant III: $\pi < \theta < \frac{3\pi}{2}$

4. Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$

Here, $ \frac{4\pi}{3} $ radians is greater than $ \pi $ and less than $ \frac{3\pi}{2}$. Hence, it lies in Quadrant III.

Given an angle of $ frac{4pi}{3} $ radians, determine the quadrant in which the terminal side of the angle lies.

To solve this:

1. Identify the standard position of the angle on the unit circle: $2pi$ radians is a full circle.

2. Since $ frac{4pi}{3} $ is more than $ pi $ but less than $ frac{3pi}{2} $, it falls in the range of Quadrant III.

Therefore, the angle $ frac{4pi}{3} $ radians lies in Quadrant III.

Given the angle $ frac{4pi}{3} $ radians, find its quadrant.

It lies in Quadrant III, as $ pi < frac{4pi}{3} < frac{3pi}{2} $.

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