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Explaining How SohCahToa Works

Date: 07/26/2004 at 14:04:33
From: Justin
Subject: Sohcahwhat

I don't do very well in math, and I'm really confused about 
trigonometry.  Can you please explain what "sohcahtoa" is all about so 
that I can understand it?  I'd really appreciate it.



Date: 07/26/2004 at 14:24:27
From: Doctor Achilles
Subject: Re: Sohcahwhat

Hi Justin,

Thanks for writing to Dr. Math.

SohCahToa is a memory trick that many students learn to help them
remember certain ratios in trigonometry.  I'll start by talking about
the ratios, then show you how SohCahToa can help you remember them.

Let's say we have a right triangle ABC with sides a, b, and c (the
side is opposite the angle and uses a small letter, so side "a" is
opposite angle "A"):

          B
         /|
        / |
     c /  |
      /   | a
     /    |
    /     |
    -------
   A   b   C

There are six possible ratios of the three sides:

  a/c   b/c   a/b   b/a   c/b   c/a

Each of these six ratios has a name in trigonometry.  Three of them
are the most commonly used, so let's look at those.

Imagine we're standing at angle A and looking at the three sides of
the triangle from there.

  The side opposite angle A is side a.

  The side adjacent (touching) angle A is side b.

  The hypotenuse of the triangle is side c (remember, the hypotenuse
  is the side opposite the right angle, and always the longest of the
  three sides).

One of the six possible side ratios is called the sine, and it's
defined as the side opposite the angle divided by the hypotenuse.  So,
from angle A:

  sine of angle A = sin(A) = opposite side/hypotenuse = a/c

The cosine ratio is defined as the side adjacent to the angle divided
by the hypotenuse.  So, from angle A:

  cosine of angle A = cos(A) = adjacent side/hypotenuse = b/c

Finally, the tangent ratio is defined as the side opposite the angle
divided by the side adjacent to the angle.  So, from angle A:

  tangent of angle A = tan(A) = opposite side/adjacent side = a/b

Now, let's look at sohcahtoa again.  Break it into three parts:

  soh cah toa

In each part, the first letter stands for the ratio name (sine, 
cosine, tangent) and the next two letters are the sides of that ratio
(opposite, adjacent, hypotenuse).

  soh means Sine = Opposite/Hypotenuse
  cah means Cosine = Adjacent/Hypotenuse
  toa means Tangent = Opposite/Adjacent

Again, sohcahtoa is just a memory trick to help you remember these
three ratios.  The other three possible side ratios are the
reciprocals of these three, and that's how people remember them.

Let's go back to our triangle again and stand at angle B instead of A:

          B
         /|
        / |
     c /  |
      /   | a
     /    |
    /     |
    -------
   A   b   C

Note that from angle B, b is the opposite side and a is the adjacent
side, while c is still the hypotenuse.  Let's use sohcahtoa to find
the three main ratios from angle B:

  soh means sin(B) = opposite/hypotenuse = b/c
  cah means cos(B) = adjacent/hypotenuse = a/c
  toa means tan(B) = opposite/adjacent = b/a

As you can see, which of the possible side ratios you get depends both
on which function (sine, cosine, tangent) you do and also on which
angle you are using.

These three basic ratios are the cornerstone of most of the work you
will do in trigonometry, so understanding them and remembering them is
important.  SohCahToa helps you do both!

If you're still stuck or you'd like to discuss some specific problems 
using trig, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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