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What Does It Mean for Angles to be Coterminal?

Date: 09/06/2007 at 18:06:35
From: Ethan
Subject: Trying to understand coterminal angles

Find one positive angle and one negative angle that are coterminal 
with an angle having measure 11 pi over 4.

I tried to understand the formula in the book but it makes no sense 
to me. Here it is: If x is the degree measure of an angle, then all 
angles of the form x + 360k degrees, where k is an integer, are 
coterminal with x.  If B is the radian measure of an angle then all 
angles of the form B + 2kpi, where k is an integer, are coterminal 
with B.

Date: 09/06/2007 at 23:32:00
From: Doctor Peterson
Subject: Re: Trying to understand coterminal angles

Hi, Ethan.

The idea of coterminal angles is that you can end up facing the same
direction after making different turns.

Let's take a simple case:

          o-------> start

Suppose you want to start facing east, and turn so that you face
north, as shown.  You could do the obvious, and turn 90 degrees
counterclockwise.  Or, you could be contrary and turn 270 degrees
clockwise.  Following the conventions of trig, we call these +90 and
-270 degrees respectively.  Note that the difference between these is
360 degrees.  Can you see why?  Together, 90 in one direction and 270
in the other make up a full 360 degrees; 90 - (-270) is 90 + 270, or
360 degrees.

Try acting these out with your body if you want; don't worry, nobody's

But you can be more creative than that!  What if you turn
counterclockwise so fast that you go past north and all the way around
again for another full turn?  You've then gone 90 degrees plus another
360, for a total of 450 degrees.  And if you enjoyed ballet, you might
do two or three or more full spins!

Or, you can spin more than once around clockwise.  We end up with all
these possible angles through which you might turn, and still end up
facing north:

  ..., -630, -270, 90, 450, 810, 1170, ...

Each one is 360 degrees--one full turn--more than the one before it.  
You're spinning a different number of times in each case.

These are called coterminal angles--meaning that they start and end
(terminate) in the same place.  They are all different turns that you
might take with the same net effect.

We can make a list of "all" of them by starting with any one 
description, say our 90 degree turn, and adding any multiple of 360 to
it (positive or negative).  A quick way to say that is

  90 + 360n degrees, where n is any integer

For example, for n = 1 we have 90 + 360(1) = 450, for n = 2 we have 90
+ 360(2) = 90 + 720 = 810, and so on for any integer n.

In terms of radians, since 90 degrees is pi/2 radians and 360 degrees
is 2pi radians, this is

  pi/2 + 2pi n radians, where n is any integer

Now, your problem is like mine, where I started with one name for the
angle, 90 degrees, and found other names (some negative, others
positive) by adding or subtracting multiples of 360 degrees.  Just
start with 11pi/4 and add or subtract 2pi (or 8pi/4) repeatedly to get
a list of possible angles.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Euclidean/Plane Geometry

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