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### 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.

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/
Associated Topics:
High School Trigonometry

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