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 seven pi over four 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 seven pi over four, the first thing that I notice is that that that numerator seven pi is larger than my denominator of four. So I know that this angle is going to be greater than one 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 three or quadrant four because looking at my unit circle starting from zero 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 three pi over two. If it is a fraction that is smaller than three pi over two, I know that it will be located in quadrant three, whereas if it's greater than three pi over two, I know that it will be located in quadrant four. So let's determine that. Now looking at this fraction three pi over two it can be really helpful to give this the same denominator as our given angle seven pi over four. So let's go ahead and do that. Now if I multiply three pi over two by two over two keeping it the same exact number but just altering that denominator. This gives me six pi over four. Now when looking at seven pi over four and six pi over four, it makes it much easier to tell which one is greater. Seven pi over four has a greater numerator with the same denominator, so I know that it is a greater angle than six pi over four. 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 seven pi over four radians is an angle located in that last quadrant, quadrant four. Thanks for watching, and let me know if you have questions.

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on 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

Verified step by step guidance

Convert the given angle $ \frac{7\pi}{4} $ radians into degrees if necessary, but since the problem is in radians, we will work directly with radians.

Recall that a full circle in radians is $ 2\pi $, and each quadrant represents a quarter of the circle: $ \frac{\pi}{2} $ radians for Quadrant I, $ \pi $ radians for Quadrant II, $ \frac{3\pi}{2} $ radians for Quadrant III, and $ 2\pi $ radians for Quadrant IV.

Determine where $ \frac{7\pi}{4} $ lies by comparing it to the quadrant boundaries: $ \frac{7\pi}{4} $ is greater than $ \frac{3\pi}{2} $ (Quadrant III boundary) but less than $ 2\pi $ (Quadrant IV boundary).

Conclude that $ \frac{7\pi}{4} $ radians is located in Quadrant IV based on its position within the interval $ \left( \frac{3\pi}{2}, 2\pi \right) $.

Verify the result by visualizing the angle on the unit circle or by subtracting $ 2\pi $ to find the equivalent angle within one full rotation, confirming it lies in Quadrant IV.