Identify the quadrant that the given angle is located in. 6 π 5 \frac{6\pi}{5} 5 6 π radians

Hey everyone. In this problem, we're asked to identify the quadrant that the given angle is located in. And here the angle we're given is six pi over five radians. Now using our knowledge of fractions here and the angles that we already know on our unit circle, we can come to a conclusion rather quickly. So this six pi over five radians, the first thing I notice is that my numerator six pi is larger than my denominator of five. So that tells me that this is a number that's definitely greater than pi. So it has to be in these bottom two quadrants, quadrant three or quadrant four. Now the other thing that we wanna look at here is how does this angle, six pi over five, relate to my angle of three pi over two? Is it less than and goes in quadrant three or greater than and goes in quadrant four? Now when looking at this, sometimes when we look at multiple fractions, we tend to wanna give them the same denominator to make them easier to look at. Now it can be kind of hard here. That would be a bit tricky to get this three pi over two to have a denominator of five so that we can directly compare these. But we can actually give them the same numerator. So let's go ahead and try this. Now if I multiply this fraction, three pi over two, by two over two in order to get a different denominator here, I can get six pi over four. Now you might be wondering why I did that because these do not have the same denominator. So how exactly is that helpful? Well, they don't have the same denominator, but they have the same exact numerator. And when fractions have the same numerator, we can tell which one is larger by which one has a smaller denominator because we're dividing it by less. So this six pi over four versus six pi over five, I know that six pi over five is going to be less than six pi over four because it has a larger denominator with the same numerator. So that tells me that my angle is less than three pi over two and has to be in between pi and three pi over two right here in quadrant three. So my angle six pi over five is going to be located in Quadrant 3. Thanks for watching and I'll see you in the next one.

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Struggling with Business Calculus?

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Multiple Choice

Identify the quadrant that the given angle is located in.
$\frac{6\pi}{5}$ radians

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Verified step by step guidance

Convert the given angle $ \frac{6\pi}{5} $ radians into degrees if necessary, but since the problem is in radians, we can proceed directly.

Recall that one full revolution around a circle is $ 2\pi $ radians, and each quadrant represents $ \frac{\pi}{2} $ radians (90 degrees).

Determine the reference angle by finding the remainder when $ \frac{6\pi}{5} $ is divided by $ 2\pi $. Since $ \frac{6\pi}{5} $ is less than $ 2\pi $, it lies within the first full revolution.

Compare $ \frac{6\pi}{5} $ to the boundaries of the quadrants: Quadrant I (0 to $ \frac{\pi}{2} $), Quadrant II ($ \frac{\pi}{2} $ to $ \pi $), Quadrant III ($ \pi $ to $ \frac{3\pi}{2} $), and Quadrant IV ($ \frac{3\pi}{2} $ to $ 2\pi $).

Since $ \frac{6\pi}{5} $ is greater than $ \pi $ but less than $ \frac{3\pi}{2} $, the angle is located in Quadrant III.

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Identify the quadrant that the given angle is located in.6π5\frac{6\pi}{5}56π radians

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Trigonometric Functions on the Unit Circle practice set

Identify the quadrant that the given angle is located in.
$\frac{6\pi}{5}$ radians