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Which Quadrant in the Unit Circle?

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 
still cannot determine the answer... please help.

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 

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 
quadrant and into the second.

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)
                          = 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 
quadrant at all!

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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