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

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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 Y express your answer in radiance. Then we're given a graph. We have a coordinate plane with a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and it's overlay with a unit circle, one of the points of which is labeled Y and we'll describe where Y is in more detail in a few minutes. Our answer choices are answer choice. A negative five pi divided by four and three pi divided by four. Answer choice B negative seven pi divided by four and three pi divided by four. Answer choice C negative five pi divided by four and two pi divided by three and answer choice D negative seven pi divided by four and two pi divided by three. All right, let's take a closer look at some of our given information to figure out what we know. All right, we've got a an angle. It's two angles in standard position, meaning the vertex is the origin and the initial side is coming from the origin and heading along the positive part of the X axis, the terminal side of our two angles, they are co terminal because the terminal side passes through point Y point Y is in the second quadrant. It's exactly halfway between the positive part of the Y axis and the negative part of the X axis. And so we can draw our terminal side passing through point Y. Now, what's the measurement of those co terminal angles in radiance? And they need to be between negative two pi and two pi. Let's talk about that for a moment. We recall from previous lessons that two pi when we're starting at the initial side, we are heading counterclockwise and going all the way an entire rotation around and ending up back where we started. That's a full rotation, that's a measurement of two pi. And we can see that our angle is within that. But if we kept going around, that would pass our limit of two pi. So for the other angle, we need to head in the negative direction. So that's also beginning at our initial side along the positive side of the X axis. But now we're heading clockwise and there we can see our angle is contained within that going all the way around would get us back to negative to pi. So to determine exactly the measurement of those two angles, we can start off by labeling the unit circle for our positive measurements. So we can start off heading counterclockwise in each quadrant, there are three hash marks and the hash marks in the first quadrant. The first one is going 1/12 of the way around our entire unit circle. So 1/12 of two pi but sometimes it's easier to think of them at least the first half in terms of pie, we know that half way around. So if we go all the way to the negative part of the X axis, that would be pi, that's a straight line, that's pi and our first hashmark in the first quadrant heading counterclockwise from the initial side is 1/6 of the way to pi. So it's a measurement of pi divided by six. The next hash mark is exactly 1/4 of the way toward that straight line toward pi. So that one is measured at pi divided by four. And the third hash mark in the first quadrant is one third of the way to pi. So that's pi divided by three. The positive part of the Y axis is halfway there. That's pi divided by two and into the second quadrant. The first hash mark, there is going to be two thirds of the way to pi. So that's two pi divided by three. And then the one we have marked the one that is where point Y is, is 3/4 of the way to pie. So that one's three pi divided by four and that's one of our angles, three pi divided by four. So now how about the negative one? Well, there are two different ways to go about doing this. One might be more intuitive to you and they're both pretty quick. So I'm going to explain them both. If the unit circle is really intuitive, then you can look at this and you can begin to label the negative values for the negative angles on our unit circle. So starting at the initial side but now heading clockwise, we have got 1/6 of the way toward now this time it's negative pie. So that's going to be negative pi divided by six for that first hashmark in the fourth quadrant as we go clockwise from our initial side, the next hash mark in the fourth quadrant is going to be a negative pi divided by four. The third hashmark in the fourth quadrant is negative pi divided by three. And you notice they're following the same pattern but they're just on the other side heading in the clockwise direction instead of counterclockwise, the negative part of the Y axis is negative pi divided by two. The first hashmark heading clockwise in quadrant three is negative two pi divided by three. The second hashmark in quadrant three is negative three pi divided by four. And then if we would have continued, we would have seen that the next hashmark here is going to be a negative five pi divided by 6 5/6 of the way toward negative pi. Now continuing after negative pie. Now it might help more in terms to think of it in terms of two pi. So in terms of a full rotation, this would simplify to be negative seven pi divided by six. That's the first hash mark in the second quadrant as we're heading clockwise and the measurement of our angle where Y is is going to be a negative five pi divided by four. All right. That is one way to come up with it. We have negative five pi divided by four and three pi divided by four. But if algebra is more intuitive for you, we know that that angle is co terminal. It's exactly one full rotation away from our three pi divided by four. And we can get it algebraically by just subtracting one full rotation, subtracting two pi from our three pi divided by four. And when we subtract those, we get negative five pi divided by four and that's a bit quicker. So looking at our answer choices, four, negative five pi divided by four and three pi divided by four that matches with answer choice. A well done, we'll catch you on the next one.

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.
D

Key Concepts

Unit Circle

Angle Measurement in Radians

Terminal Side of an Angle

Watch next

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.D

Key Concepts

Unit Circle

Angle Measurement in Radians

Terminal Side of an Angle

Watch next

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.
D