Identify the quadrant that the given angle is located in. 2 π 3 \frac{2\pi}{3} 3 2 π radians

In this problem, we're asked to identify what quadrant of our unit circle our given angle is in and the angle we're given here is in radians. It's 2 pi over 3 radians. Now again, here we're going to use our knowledge of fractions and those angles that we already know on our unit circles. So looking at this angle, 2 pi over 3. Something that might be helpful to you when looking at problems specifically like this is to ignore that pi in your radian angle measure. Now you shouldn't do this all the time, but whenever we're just looking at where our angle fits into our unit circle, this might be helpful because you might be more familiar with fractions when you don't have to worry about that pie. So here, if we consider this angle or this fraction to be 2 thirds and we look at our unit circle and we see that we start with fraction to be 2 thirds and we look at our unit circle and we see that we start with 0 and ignoring this pie here, this is really a fraction of 1 half. And then here with this pie, this is a fraction of 1. Now when we consider where 2 thirds fits into that, I know that 2 thirds fraction of 1. Now when we consider where 2 thirds fits into that, I know that 2 thirds is less than 1, so I know that it has to be either in quadrant 1 or quadrant 2. Now when I consider its relation to 1 half, 2 thirds is a fraction that is definitely larger than 1 half. So I know that 2 thirds is greater than 1 half, which also tells me that 2 pi over 3 is greater than pi over 2. So that tells me that this fraction, this angle, sits right in between my fraction of pi over 2 and pi right in quadrant 2. So this angle, 2 pi over 3 radians, is in quadrant 2. Thanks for watching, and let me know if you have questions.

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9. Unit Circle

Defining the Unit Circle

Precalculus

9. Unit Circle

Defining the Unit Circle

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Multiple Choice

Identify the quadrant that the given angle is located in.
$\frac{2\pi}{3}$ radians

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

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Verified step by step guidance

Convert the given angle from radians to degrees to make it easier to understand. The conversion formula is: degrees = radians × (180/π).

Apply the conversion formula to the given angle $ \frac{2\pi}{3} $ radians: degrees = $ \frac{2\pi}{3} \times \frac{180}{\pi} $.

Simplify the expression by canceling out $ \pi $ and multiplying the remaining numbers to find the degree measure of the angle.

Determine the quadrant by analyzing the degree measure. Quadrant I is 0° to 90°, Quadrant II is 90° to 180°, Quadrant III is 180° to 270°, and Quadrant IV is 270° to 360°.

Since the degree measure of $ \frac{2\pi}{3} $ radians falls between 90° and 180°, conclude that the angle is located in Quadrant II.