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 are 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 K express your answer in radiance. Then we are given a graph. It's a coordinate plane. We have a vertical y axis, a horizontal X axis. They come together at the origin in the middle. And then on the positive part of the y axis, we have our point K. Our answer choices are answer choice. A negative two pi divided by three and pi divided by four. Answer choice B negative two pi divided by three and three pi divided by two answer choice C negative three pi divided by two and pi divided by four and answer choice D negative three pi divided by two and pi divided by two. All right. So what do we know about these two angles? And that we need to express in radiance? Well, we've got a standard position couple of angles here. So that means that its initial side begins at the origin, has its vertex at the origin and extends through the positive part of the X axis. Now how about the terminal side? We're told that that starts at the origin and passes through point K. So that means from the origin, the terminal side is heading up along the positive part of the y axis through our point K. So that is our angle, but we need to express it as two different angles. And then our restrictions are, is that the two angles need to be between negative two pi and positive two pi. So we recall from previous lessons that two pi is when our angle begins at the positive part of the X axis. And then it heads counterclockwise and two pie means that it's gone all the way around right back to where it started now for negative two pi. That means we've got our angle that starts at the positive part of the X axis. And this one is heading clockwise all the way around until it gets back to where it started at the X axis. That is our negative two pie. So our angles need to be between that. So what's the measurement for this angle when we're talking about our positive two pie? Well, we know that all the way around is two pi we also know that halfway around is positive pie and halfway around half or in other words, one quarter of, of his multiply of the way of the whole way around one quarter of two pi is two pi divided by four, which is the same as pi divided by two. So the positive part of the y axis that is pi divided by two. And that is one measure of this angle. A positive pi divided by two that is between negative two pi and two pi. Now how about if we go the other way around we take the negative angle. So in this case, we go halfway around. Well, that's negative pi but what if we go another half? So we're taking our pie plus another half of pie. Well, pi if we have a common denominator, that would be two pi divided by two, adding these together, we'd get three pi divided by two. But remembering that when we're heading in that clockwise direction, these are negative. So that's a negative. The pi divided by two, that's our other angle. And another way that you could find it is we know we had pi divided by two. Well, you can subtract that full rotation from it. You can subtract two pi and when you take pi divided by two minus two, pi you again get negative three pi divided by two. So we look at our answer choices and that matches with answer choice. D 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.
E

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.E

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.
E