Use the circle shown in the rectangular coordinate system to solv...

Question

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.
A

Circle in rectangular coordinates showing angles in standard position for trigonometry exercises.

A

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine two angles between negative two pi and two pi where the terminal side of each angle passes through the origin and point X express your answer in radiance. Then we're given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and they're overlain with a unit circle on the unit circle. One of the points is marked X. We'll describe where in more detail in a couple moments. Our answer choices are answer choice. A negative nine pi divided by 11 and pi divided by four answer choice B negative seven pi divided by six and pi divided by six. Answer choice C negative 11 pi divided by six and pi divided by six and answer choice D negative pi divided by 11 and pi divided by three. All right. Let's take a look at what we know. We know we have an angle well, two angles in standard position. And that means that the initial side is going to have the vertex of the origin heading toward the positive part of the X axis the terminal side of our two angles, they co terminal because starts at the origin and the terminal side passes through X X is in the first quadrant. And on our unit circle, it's on the first hash mark that is one third of the way between the X axis and the Y axis. Closer to the X axis, the positive parts of both because it's in quadrant one. So we can draw our terminal side passing through point X oh close. It's almost there. And what we need to determine is what are the measurements of those two co terminal angles. And there are several ways to do this. Um We know that we're limited because our co terminal angles need to be between negative two pi and two pi. Let's talk about what that means a little bit if we start from our initial side and if we were to go clock or excuse me, counterclockwise, heading up and going all the way around and coming back to where we started, that's a measurement of two pi one complete rotation is two pi. And we can see that as we were heading counterclockwise, our angle would have been within that. But if we were to keep going around and then head to that angle one more time, we would have passed two pi. So the only other way to remain between negative two pie and two pie is now for us to go in the negative direction from our initial side, if we were to go clockwise heading all the way around in a complete rotation from the initial side, that's a measurement of negative two pi and we can see that our angle was within that as well. So now we just need to figure out what that exact measurement is and it helps if we label our unit circle. So we'll start off finding that positive measurement of the angle. And from our initial side heading counterclockwise into the first quadrant. As we said, that first mark is one third of the way through the first quadrant. Another way to think of it is that it is 1/6 of the way toward a straight line. And the measurement of a straight line we recall from previous lessons is pi that's half of two pi it's pi that's the negative part of the X axis. So that first measurement is 1/6 of pi and we would label that as pi divided by six and there is our first angles measurement, the positive one. Now, how about the negative one? Well, we could likewise label the unit circle and this might be more intuitive for you or you might need the practice labeling the unit circle. And so what you do with that is you start on the terminal or excuse me, on the initial side, the positive part of the X axis. And now instead you head clockwise down into the fourth quadrant and this time they're negative angles. And this one is 1/6 of the way toward a straight line. But a straight line here is the negative angle. It's negative pi. So 1/6 of negative pi is negative pi divided by six. That's the first hash mark in the fourth quadrant heading clockwise from the initial side. Now the next hash mark in the fourth quadrant, that one is 1/4 of the way toward negative pi. So that's negative pi divided by four. And the next hash mark in the fourth quadrant, that's one third of the way to negative pies. That's negative pi divided by three and the negative part of the Y axis that's halfway to negative pi. So that's negative pi divided by two. Now we can continue to label this and I will so that you've got the information. But if you need a refresher on labeling the unit circle, definitely check out previous lesson videos. So I'm going to go more quickly as we label the third for 2nd and 1st quadrants. So heading clockwise, the first hash mark in the third quadrant is negative pi or excuse me, negative two pi divided by three second hash mark in the third quadrant is negative three pi divided by four. The third hashmark in the third quadrant is negative five pi divided by six. Then we have negative pi which is the negative part of the X axis. Now into the second quadrant continuing clockwise, we have negative seven pi divided by six. Then we have negative five pi divided by four. And our last mark in the second quadrant is negative four pi divided by three. The positive part of the Y axis is negative three pi divided by two. And then into the first quadrant continuing clockwise, we have negative five pi divided by three. Then negative seven pi divided by four. And finally where our X is that is negative 11 pi divided by six. So there's our other angle measurement for those co terminal angles negative 11 pi divided by six. Now there is a faster way to do this. If algebra is more intuitive to you, we know that from our measurement of pi divided by six, if we had one full rotation in the negative direction, we arrive back at that angle. And so one full rotation in the negative direction is negative two pi. So we can take pi divided by six minus two pi and that will equal negative 11 pi divided by six. So whichever of those two is more helpful to you? Definitely use that or practice both. If you need practice, we look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.