Identify the quadrant that the given angle is located in. 7 π 4 \frac{7\pi}{4} 4 7 π radians

Hey everyone. In this problem, we're asked to identify the quadrant that the given angle is located in. And the angle that we're given is in radians. It's 7 pi over 4 radians. Now looking at this problem, you can choose to convert this radian angle measure into degrees if that makes it more simple for you. But here, I'm going to walk you through how to identify where this angle is located using exclusively its radian measure just so that we get a little bit more familiar with radians along the way. So let's go ahead and get started. Now looking at this angle 7 pi over 4, the first thing that I notice is that that numerator 7 pi is larger than my denominator of 4. So I know that this angle is going to be greater than 1 or in this case, we can look at it in relation to pi. This is going to be an angle that is a greater than pi. So I know that this has to be located either in quadrant 3 or quadrant 4 because looking at my unit circle starting from 0 going around, here is pi, so I know that it has to be in that bottom in those bottom two quadrants. So now we need to look at its relation to this other angle, 3 pi over 2. If it is a fraction that is smaller than 3 pi over 2, I know that it will be located in quadrant 3. Whereas if it's greater than 3 pi over 2, I know that it will be located in quadrant 4. So let's determine that. Now looking at this fraction 3 pi over 2, it can be really helpful to give this the same denominator as our given angle 7 pi over 4. So let's go ahead and do that. Now if I multiply 3 pi over 2 by 2 over 2, keeping it the same exact number but just altering that denominator, This gives me 6 pi over 4. Now when looking at 7pi over 4 and 6pi over 4, it makes it much easier to tell which one is greater. 7pi over 4 has a greater numerator with the same denominator, so I know that it is a greater angle than 6pi over 4. Now because of that, that tells me that my angle will be located to the right of this going clock counterclockwise around my circle. So that tells me that 7 pi over 4 radians is an angle located in that last quadrant, quadrant 4. Thanks for watching, and let me know if you have questions.

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Precalculus

9. Unit Circle

Defining the Unit Circle

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Multiple Choice

Identify the quadrant that the given angle is located in.
$\frac{7\pi}{4}$ 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 if necessary, but in this case, we will work directly with radians.

Recall that the unit circle is divided into four quadrants, each covering an angle of π/2 radians (or 90 degrees).

Determine the reference angle by finding the equivalent angle between 0 and 2π radians. Since 7π/4 is already between 0 and 2π, we can use it directly.

Identify the quadrant by comparing the angle to the boundaries of each quadrant: Quadrant I (0 to π/2), Quadrant II (π/2 to π), Quadrant III (π to 3π/2), and Quadrant IV (3π/2 to 2π).

Since 7π/4 is greater than 3π/2 and less than 2π, it falls in Quadrant IV.