Trigonometric Identities

Date: 11/12/97 at 18:16:50
From: Steven Schoenherr
Subject: Trigonometric Identities

I have a huge test coming up on trig identities and was wondering what 
the best way to solve verification problems would be.  Also, do I 
really have to memorize all of the Identities in order to understand 
them?
                                                                                
Thanks,
Steven Schoenherr

Date: 11/12/97 at 21:00:46
From: Doctor Wilkinson
Subject: Re: Trigonometric Identities

There are some identities that you have to know, and others that you 
should be able to derive from the ones you have memorized.

I divide the essential identities into three groups:

(1) Identities that express all the trig functions in terms of the 
    sine and cosine:

    tanx = sinx/cosx
    cotx = cosx/sinx
    secx = 1/cosx
    cscx = 1/sinx

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

This is just the Pythagorean Theorem expressed in terms of 
trigonometry.

Using (1) and (2) you can prove a whole bunch of other identities, and 
this saves you having to memorize them.  For example,

 sec^2x - tan^2x =
    1/cos^2x - sin^2/cos^2x   (from group 1)
     =  (1 - sin^2x)/cos^2x
     =  cos^2x/cos^2x         (from 2)
     =  1

(3) The addition formulas for sine and cosine:

    sin(x + y) = sin(x) cos(y) + cos(x) sin(y)    and

    cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

From the formulas in (3) and the formulas in (1) you can derive 
addition formulas for the other functions such as the tan. You can 
also get the so-called double-angle formulas like

    sin(2x) = 2sin(x) cos(x)

I also recommend doing a lot of practice problems, but keep the 
memorization down to just the identities I've mentioned already.

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