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.

16𝜋 3

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 given is 21 pi divided by four. We are provided a blank unit circle to begin with. I recall that the full distance around my unit circle is too high. Looking at the angle we're given 21 pi divided by four that is definitely greater than two pie. So to sketch this in standard position, I want to get this back in a standard form by subtracting two pie from 21 pi divided by four. And when I make a common denominator, I have 21 pi divided by four minus eight pi divided by four. And this will give me 13 pie divided by four, which is definitely still greater than two pie. Since we now know two pie is equivalent to eight pie divided by four. So we'll subtract another rotation. So another eight pi divided by four. And when we subtract this, we get five pi divided by four as the angle we're working with. So now to see what quadrant this falls into, I'm gonna start by making one side of my angle on the positive side of the x axis because in standard form, that's where one side starts. And we need to find where the other side ends. Recalling that the positive side of the Y axis or the distance from zero to the vertical line is pi divided by two. And then the distance from the positive side of the X axis all the way around the unit circle to the negative side of the X axis is pi I can see if five pi divided by four falls in either quadrant one or quadrant two. And if it doesn't, I will keep finding values to see how they fit. So pi divided by two, if I were to make a common denominator of four would be two pi divided by four. So our angle is definitely not in the first quadrant pie would be equivalent to four pi divided by four. So our angle is definitely not in the second quadrant. So the pos uh the negative side of the y axis would be three pi divided by two. So if I make this equivalent to a denominator of four, I'd have six pi divided by four. So our angle of five pi divided by four would definitely fall in the third quadrant as it is between four pi divided by four and six pi divided by four. So we know this is quadrant three. Now to graph it five pi divided by four is the precise middle of four pi divided by four and six pi divided by four. So it continues to this middle line in quadrant three on our unit circle and this is where our angle would fall. So it's in quadrant three and it goes to the middle line on the unit circle. And this full angle here is five pi divided by four, have a nice day.