Derivation of Sum/Difference of Sine, Cosine, Tangent

Date: 02/16/2002 at 22:32:24
From: Meredith
Subject: Derivation of the Sum/Difference of Sine, Cosine, and Tangent

How can I find the derivation of the sum/difference of sine, cosine, 
and tangent?

Date: 02/17/2002 at 00:07:41
From: Doctor Schwa
Subject: Re: Derivation of the Sum/Difference of Sine, Cosine, and 
Tangent

Hi Meredith,

A beautiful geometric approach is at

   Geometry of Addition and Subtraction Formulas - Smiley
   http://www.math.uaa.alaska.edu/~smiley/trigproofs.html   

and another approach, which I got from John Conway, is at

   Trig/Analyt Handouts - Joshua Zucker
   http://www.castilleja.org/faculty/josh_zucker/gunn/trig/handouts/   
   (select the link to "a derivation of the angle addition formula 
    for sine - PDF format)

A very nice exposition of the standard geometric proof is in our 
archives:

   Cosine Addition Formula
   http://mathforum.org/dr.math/problems/memon12.13.97.html   

with a slightly different approach at

   Sine and Cosine Addition Formulas
   http://mathforum.org/dr.math/problems/phara.11.29.00.html   

There's also the derivation using complex numbers, which is
suggested at

   Trigonometric Functions and Euler's Identity
   http://mathforum.org/dr.math/problems/jungbok4.10.98.html   

and made much more explicit at

   Euler's Formula
   http://mathforum.org/dr.math/problems/wolf1.27.98.html   

I hope at least one of those methods appeals to you.  Let me know
if you would like more information about any of them.  My favorites
are Conway's and the complex number method in that last link.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/