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

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 two pi divided by three. We are provided a blank unit circle. So the first thing I want to consider when looking at the angle, two pi divided by three is is this greater than one rotation around our unit circle? Well, one full rotation is two pi. So if this is two pi divided by three, it has to be less than one full rotation because it's a full rotation divided in thirds. So now I will graph it or sketch it on my unit circle. I recall that when I'm in standard position, one side of my angle is on the positive side of the X axis. And the other side is gonna be where the angle lands the quadrant that we're looking for. So first, I'm gonna label ingredients my axes that are on this unit circle. So the positive side of the x axis is zero. The positive side of the y axis is pi divided by two. The negative side of the X axis is pi and the negative side of the y axis is three pi divided by two. I do this. So I can try to find out where on this unit circle, the angle will land looking first at pi. And if I want to write that with a common denominator to my angle, two pi divided by three, I would use it as three pi divided by three. And since two pi divided by three is less than three pi divided by three, I know that our given angle is between zero and pi. So it is not in quadrants three or four. Now, if I want to compare it to pi divided by two, I need to make a common denominator. In this case between two and three, my common denominator would be six. So pi divided by two would be equivalent to three pi divided by six and two pi divided by three would be equivalent to four pi divided by six. So this means that this is greater than two pi or than pi divided by two, but less than pie. So it needs to fall in the second quadrant. So two pi divided by three is in quadrant two. Next, I need to find the angle that we actually have. So I need to find out where does it line up. And if you wanted to reference another unit circle, you would find that this first line in quadrant two is exactly two pi divided by three. So on our unit circle, this is what the angle two pi divided by three would look like it would land right here in the second quadrant. Have a nice day.

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.

3𝜋/4

Key Concepts

Angles in Standard Position

Radian Measure and the Unit Circle

Quadrants of the Coordinate Plane

Watch next

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.3𝜋/4

Key Concepts

Angles in Standard Position

Radian Measure and the Unit Circle

Quadrants of the Coordinate Plane

Watch next

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.

3𝜋/4