Finding Intervals in Trig Graphs

Date: 10/25/98 at 18:11:30
From: sara
Subject: Graphing sine and cosine functions

I'm in pre-calculus, and I don't understand graphing sine and cosine 
functions. I understand it up until you have to divide the interval 
into four equal parts. One of the problems I'm having a hard time with 
is the interval negative pi divided by four to seven pi divided by 
four. In the book, you have to add the two together and multiply by 
1/4, 1/2, and 3/4. When I add the two intervals, I get six pi over 
four. Then I multiply it by 1/4, and I get three pi over eight. But 
the answer is pi over four. Could you please help me?

Date: 10/26/98 at 12:30:44
From: Doctor Peterson
Subject: Re: Graphing sine and cosine functions

Hi, Sara. I think what you are trying to do is to divide the interval:

   (-pi/4, 7pi/4)

into four equal parts. The fact that it doesn't start at zero is what 
makes this difficult for you, together with the fact that the left end 
is negative. I get the impression you are just trying to follow an 
example that did start at zero, without thinking about what it means. 
So let's start out by picturing what we're doing, rather than just 
working with numbers.

Here's your interval:

   ---+===+===+===+===+===+===+===+===+---
     -pi  0               pi         7pi
     ---                             ---
      4                               4

First we have to measure how long the interval is. If you trust my 
drawing, you can just count the eight pi/4 marks I've made. If you 
don't, what you have to do is to subtract (NOT ADD!) the two ends:

   7 pi    -pi   (7 - -1) pi   8 pi
   ---- - ---- = ----------- = ---- = 2 pi
     4      4         4          4

Now if we want to divide this interval into four equal parts, each one 
will be a quarter of this length:

   2 pi   pi
   ---- = --
     4     2

Now the points where you have to "cut" the interval will be every pi/2, 
starting at your left end point:

   -pi   pi   pi
   --- + -- = --
    4     2    4

   pi   pi   3 pi
   -- + -- = ----
    4    2     4

   3 pi   pi   5 pi
   ---- + -- = ----
     4     2     4

Just as a check, the next point should be your right end point:

   5 pi   pi   7 pi
   ---- + -- = ----      looks good!
     4     2     4

So the 1/4, 1/2, and 3/4 points are pi/4, 3pi/4, and 5pi/4:

   ---+===+===+===+===+===+===+===+===+---
     -pi  0   pi     3pi     5pi     7pi
     ---     ---     ---     ---     ---
      4       4       4       4       4

The two mistakes you made were adding rather than subtracting, and then 
stopping too soon and taking the length of 1/4 of the interval as the 
answer, without adding it to the starting point.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/