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Trigonometric Identities

Date: 05/24/99 at 22:51:25
From: Christy Miller
Subject: Trigonometric Identities

I just started in Trigonometry with Identities. (I will use @ to mean 
theta.) I understand how to verify or prove basic identities true such 
as: sec@ = 1/cos@, but I do not understand how to simplify problems 
such as 

   sin^2(@) - 1

One example the book gives is that sec^2(@) - 1 simplifies to 
tan^2(@). I do not see at all where they can get that answer.

One other question I had was; where does theta come from? Does it 
always represent the angle on the x/y axis of a Cartesian plane?

Thank you so much for your help.

Date: 05/25/99 at 11:05:17
From: Doctor Rick
Subject: Re: Trigonometric Identities

Hi, Christy, welcome to Ask Dr. Math.

The first example you give of an identity comes straight from the
definitions of secant and cosine in terms of the unit circle. This 
kind of identity is quite important, but another kind comes from 
perhaps the most widely used theorem in math - the Pythagorean 

Take a look at the unit circle:

           *             O
        *               /|  *
      *                / |    *
    *               1 /  |      *
   *                 /   |sin@   *
   *                /    |       *
  *                / @   |        *
  *                 cos@          *
   *                             *
   *                             *
    *                           *
      *                       *
        *                   *
           *             *

You can see from the right triangle that the Pythagorean theorem, in
trigonometric terms, is

  sin^2(@) + cos^2(@) = 1

This fundamental trigonometric identity is VERY useful in proving 
other identities. Watch how it is used to simplify sec^2(@)-1:

  sec^2(@) - 1 = -------- - 1

                    1       cos^2(@)
               = -------- - --------
                 cos^2(@)   cos^2(@)

                    1 - cos^2(@)
               = ---------------

                 sin^2(@)    (Pythagoras!)
               = --------

               = tan^2(@)

>One other question I had was; where does Theta come from? Does it 
>always represent the angle on the x/y axis of a Cartesian plane?

You can always think of the argument of a trig function (theta) as the 
angle in the unit circle, whenever you need to go back to definitions 
to understand a trig concept. But theta does not have to be the angle 
with the x axis, or even any angle that you can measure. The trig 
concepts that germinate in the unit circle can be transplanted to all 
sorts of environments. 

I work with digital (high definition) television. Part of the process 
of encoding pictures digitally for transmission to your home is 
something called the Discrete Cosine Transformation. This cosine 
doesn't measure any angle you can see in the picture; rather, it has 
to do with how the brightness of the picture varies as you move across 
the picture. 

You'd be amazed at the places that trigonometry shows up!

- Doctor Rick, The Math Forum   
Associated Topics:
High School Trigonometry

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