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. We are asked for the given angle 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 negative three pi divided by four. And we are provided a blank unit circle. First recall that if the angle is negative, this means that in standard position, we are starting at the positive side of the X axis and instead of going up or counterclockwise, we're gonna go down and clockwise. Another way we can look at this, however is by creating a reference angle and I can create a reference angle by adding two pie to my angle. And that's what I'm gonna do because I rather work in positives than negatives. So negative three pi divided by four plus two pie, which we would rewrite as eight pi divided by four is five pi divided by four. So this is my reference angle. I'm gonna draw a line where five pi divided by four falls. But I'm gonna make sure I label my angle negative three pi divided by four. So looking for five pi divided by four on my unit circle, I'm gonna label ingredients, the axes. So the positive side of X is zero or two pi, the positive side of Y is pi divided by two. The negative side of X is pi and the negative side of Y is three pi divided by two. So using my reference angle of five pi divided by four, I'm gonna start by looking at pie. If I rewrite that with a common denominator of four pi divided by four, I know that five pi divided by four is greater than that. So it will not be in quadrant one or two. I am almost positive. This will land in quadrant three but I do want to double check. So I'm gonna make a common denominator from three pi divided by two. She gives me six pi divided by four and five pi divided by four will fall in between four pi divided by four and six pi divided by four. So this means that the angle negative three pi divided by four is in quadrant three. Now, as it is the exact middle of our two axes, we can draw it directly to the middle. And so this is our direct line on our unit circle. You also could reference a unit circle that has all the other information on it. And when I label it, I'm gonna make sure I label down and clockwise and label it as negative three pi divided by four and not as my reference angle. So we have an angle that starts at the x axis and goes down to the third quadrant and that's negative. Three pi divided by four 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.

-2𝜋/3

Key Concepts

Angles in Standard Position

Radian Measure

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.-2𝜋/3

Key Concepts

Angles in Standard Position

Radian Measure

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.

-2𝜋/3