Signs of Sines (and Other Trigonometric Ratios)

Date: 07/11/2002 at 10:17:44
From: Amul
Subject: Signs of trigonometric ratios

Dear Dr. Math,

The trigonometric ratios have the signs + or -, depending on the
quadrant. How do we know it is "+" or "-" among four quadrants? 

Thank you.

Date: 07/12/2002 at 09:17:23
From: Doctor Peterson
Subject: Re: Signs of trigonometric ratios

Hi, Amul.

It's easy to keep track of the signs if you think about the 
definition of the functions in terms of a unit circle. Picture the 
point (x,y) moving around the unit circle; x and y change sign as you 
move from quadrant to quadrant, and the ratios take their signs from 
the combinations of these:

                          sin   cos   tan
    quadrant  x   y   r   y/r   x/r   y/x
    -------- --- --- ---  ---   ---   ---
        I     +   +   +    +     +     +
       II     -   +   +    +     -     -
      III     -   -   +    -     -     +
       IV     +   -   +    -     +     -

You can picture it this way:

       cos -     cos +

              |
              |
       tan -  |  tan +    cos +
              |
            II|I
    ----------+----------
           III|IV
              |
       tan +  |  tan -    sin -
              |
              |

That is, the sine is positive on the top (when y is positive), the 
cosine is positive on the right (when x is positive), and the tangent 
is positive when the sine and cosine have the same sign.

This may help you, too:

      http://mathforum.org/library/drmath/view/53943.html 

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

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