In Exercises 41–56, use the circle shown in the rectangular coord...

Question

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
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Circle in rectangular coordinates for measuring angles in standard position.

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Accepted Answer

Hello everyone. For the angle given below, we are asked to identify the quadrant where it lies and sketch it in standard position on the unit circle provided the angle we are working with is negative four pi divided by three. We are given a blank unit circle to work with. First thing I notice is that my angle is negative. So this means that when we're in standard position, we have one side that will be on the positive side of the X axis and we will be going down and clockwise to find the other side. However, I would rather work with a reference angle that is positive. So to create a positive reference angle, I'm gonna add two pi which is one full rotation to my negative angle. So I have negative four pi divided by three plus six pi divided by three. And so my positive reference angle will be two pi divided by three. I do this because I work easier in positive numbers than negative numbers. So I would rather do that and be able to flip back and forth. So marking my X and Y intercept in terms of their radiant values, the positive side of the X is zero or two pi the positive side of the Y is pi divided by two. The negative side of the X is pi, the negative side of the Y is three pi divided by two. So now to see where this falls, well, if I think about fractions like two thirds, I know two thirds is less than one pie is one pie. So this means that it has to be between zero and pi. So in quadrants one or two, we won't be graphing this where it ends in quadrant three or four because it will not end in quadrant three or four to find out if it falls in quadrant one or two, I'm going to use a common denominator. So I'm gonna say two pi divided by three is the same as four pi divided by six. And I did that. So I can make my ra my radiant measure that I have at the top of the Y axis into three pi divided by six. This way I can see which one's bigger. Well, four pie is larger than three pie. So I know that this will be in quadrant two. So even our negative angle negative four pi divided by three is in quadrant two. And if you reference a unit circle, you will find that right here. The first line in quadrant two is the location of two pi divided by three. So this means that our angle ends in quadrant two on that first line. However, again, recall we're working with a negative angle. So we start at the axis, the X axis and we go count clockwise and that's gonna be the measure for negative four pi divided by three. It lands in quadrant two and it will end on the first line. If you're going to the left of the, of the y axis, have a nice day.