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 6 pi over 5 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 6 pi over 5 radians, the first thing I notice is that my numerator 6 pi is larger than my denominator of 5. 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 3 or quadrant 4. Now the other thing that we wanna look at here is how does this angle, 6 pi over 5, relate to my angle of 3 pi over 2? Is it less than and goes in quadrant 3 or greater than and goes in quadrant 4? 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 3 pi over 2 to have a denominator of 5 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, 3pi over 2, by 2 over 2 in order to get a different denominator here, I can get 6pi over 4. 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 6 pi over 4 versus 6 pi over 5, I know that 6 pi over 5 is going to be less than 6 pi over 4 because it has a larger denominator with the same numerator. So that tells me that my angle is less than 3pi over 2 and has to be in between pi and 3pi over 2 right here in quadrant 3. So my angle 6pi over 5 is going to be located in quadrant 3. Thanks for watching, and I'll see you in the next one.

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Precalculus

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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

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Verified step by step guidance

Convert the given angle from radians to degrees to make it easier to understand. Use the conversion factor: 180 degrees = π radians. Therefore, multiply $ \frac{6\pi}{5} $ by $ \frac{180}{\pi} $ to convert it to degrees.

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

Determine the equivalent angle within the standard 0 to 360-degree range by subtracting 360 degrees if necessary. This helps in identifying the correct quadrant.

Recall the quadrant rules: Quadrant I (0 to 90 degrees), Quadrant II (90 to 180 degrees), Quadrant III (180 to 270 degrees), and Quadrant IV (270 to 360 degrees).

Compare the calculated degree measure of the angle to these ranges to identify that the angle $ \frac{6\pi}{5} $ radians is located in Quadrant III.