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### Which Quadrant in the Unit Circle?

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Date: 11/30/98 at 21:57:23
From: Rosalynn
Subject: The Unit Circle

Hello, I'm trying to determine how to do this problem:

"for each value of s, find the quadrant in which C(s) is found."

I've studied the examples in my textbook. One example was:

Find the quadrant in which C(s) is located

s = 14pi/3
= 2pi/3 + 4pi
= 2pi/3 + 2(2pi)

C(14pi/3)= C(2pi/3)

Thus, C(14pi/3) is in quadrant II

I tried to apply this example to one of my problems, s=15pi/3, but I
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Date: 12/01/98 at 13:30:35
From: Doctor Peterson
Subject: Re: The Unit Circle

Hi, Rosalynn.

I assume your "C(s)" is the point on the unit circle at angle (or arc
length) s from the point (1,0). Here's what you need to remember to do
this:

Each quadrant takes up pi/2 radians; that is, for every pi/2 that s
increases, you go into a new quadrant.

After four quadrants (4 * pi/2 = 2 pi), you come back where you
started, since you've gone all the way around the circle.

So what I would do is to divide s by 2 pi to get the number of times
you went around the whole circle, and keep the remainder; then multiply
the result by 4 to find the quadrant number.

For example, for 14pi/3, we divide by 2 pi:

14 pi
-----
3      14 pi    1     7
------- = ----- * --- = --- = 2 1/3 circles
2 pi      3     2pi    3

Two full circles takes us back to the starting point; how many
quadrants does 1/3 circle take us through? Multiply by 4, and we get
4/3 = 1 1/3 quadrants. So we've gone entirely through the first

In terms of the explanation in your book, we can rewrite our division
as a multiplication:

14 pi
----- = (2 1/3)(2 pi) = 2 (2 pi) + (1/3) (2 pi)
3
= 2 (2 pi) + 2/3 pi
= 2 (2 pi) + (4/3) (pi/2)
= 2 circles + 1 quadrant + a little more

Your problem, 15/3 pi = 5 pi, will be a little special:

5 pi     5
------ = --- = 2 1/2 circles
2 pi     2

This takes us through exactly 2 full circles and 2 more quadrants, so
we are right on the line between quadrants II and III. We're not in a

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Conic Sections/Circles
High School Geometry

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