Trig Functions

Date: 9/29/96 at 16:19:20
From: Ron Turner
Subject: Help

Dr. Math,

I am having a real problem knowing when to use sine, cosine, and 
tangent.  I am taking geometry by home study and don't quite 
understand.  

Thanks!
Shauna

Date: 10/29/96 at 20:16:16
From: Doctor Lynn
Subject: Re: Help

Shauna,

The first thing to remember is that sine and cosine are opposites, but 
tangent is different because it doesn't use the hypotenuse.

In a triangle that looks something like the following, the three 
definitions are:

sin x = Opposite/Hypotenuse
cos x = Adjacent/Hypotenuse
tan x = Opposite/Adjacent

              /|
             / |
            /  | 
           /   |
          /    | 
         /     |
     H  /      |
       /       | O
      /        |
     /         |
    /          |   Some people like to remember the rules like this: 
   /           |       
  / \          |         O       A       O
 / x \         |       -----   -----   -----   (Soh-Cah-Toa)
/--------------|       S * H   C * H   T * A
     A

To use the triangles of letters, you cover up what you want to find.

So to find the formula for sine, you cover up the S in the first 
triangle and you're left with O/H = opposite/hypotenuse.  To find a 
formula for the hypotenuse, you cover the H in either the first or the 
second triangle and you're left with either O/S = opposite/sine or 
A/C = adjacent/cosine.

As an example, say you want to find the height of this triangle:

              /|
            /  |
   10 m   /    |
        /      |  
      /        |
    /          | 
  /  30\       |
/--------------|

This means that you are trying to find the length of the side which is 
opposite the 30 degree angle, so you could use either sine or tangent.  
You only know the length of the hypotenuse (10 m), however, so you 
can't use the tangent.  So you use the first triangle of letters:

    O
  -----
  S * H

When you cover up the O (what you want to find), this becomes:

  S * H

The opposite side = Sine * Hypotenuse = sin 30 * 10m = 0.5 * 10m = 5m.
So the triangle is 5m high.

If you remembered the three definitions, you could have used those 
too:

    sine = opposite/hypotenuse
  sin 30 = opposite/10
opposite = 10*sin 30 = 10*0.5 = 5m

One more point - you can remember the letter triangles with this 
mnemonic:

  Silly Old Hag Cracks All Her Teeth On Apples.
  S  =  O / H,  C =    A / H,  T  =  O/ A

The only way to get really good at trigonometry is practice :)

I hope that helps.  Please write back if you have any more problems.

-Doctor Lynn,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/