Angles as Turns

Date: 05/29/2003 at 10:05:27
From: Jimmy
Subject: Angles

How can angles be negative?

I don't know what negative angles are.

Date: 05/29/2003 at 12:46:36
From: Doctor Peterson
Subject: Re: Angles

Hi, Jimmy.

This is similar to asking how a distance can be negative.

Normally we think of distances as lengths, which must be positive. 
But sometimes we are thinking of directed distances, such as how far 
we are above sea level; then a distance below sea level would be 
"backward," and would be given a negative sign.

Similarly, in geometry we normally think of an angle as just the
"distance" between two rays, and it is always positive. But 
(particularly in trigonometry) we can also think of an angle as a 
rotation, which can go either clockwise or counterclockwise, so the 
direction counts. Just as we think of "up" as positive, someone 
decided long ago to think of a counterclockwise turn as positive 
(probably because an angle in standard position, with the starting 
ray along the x axis, starts out going up when you go 
counterclockwise). So a counterclockwise angle is positive, and a 
clockwise angle is negative:

         /
        /  up
       /    ^
      /     |
     /+     |          clockwise
    +----------        +----------
    counterclockwise    \-     |
                         \     |
                          \    v
                           \  down
                            \

When the angle is not in standard position, we still think of a 
clockwise turn as negative.

So here's the main point: when we think of an angle just as a figure 
on the paper, and want to measure how big it is, the answer is always 
positive. But when we think of an angle as a turn, starting at one 
ray and ending at another, then we will include a sign in the answer, 
showing which direction it turns.

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

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