Trigonometric Functions and Euler's Identity

Date: 04/10/98 at 03:12:03
From: Seo Jung-bok
Subject: trigonometric function

Hi, Dr. Math.

We are high school students of 'Pusan Highschool of Science' in Korea.
We have some questions. We have to do research about trigonometry 
function and PROVE some formulas.

We know some answers, but we want something special.
Please answer in VARIOUS WAYS, as MANY as possible.

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

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

   tan(x+y)=[tan(x)+tan(y)]/[1-tan(x)tan(y)]

Thank you for your help.
Bye.

Date: 04/11/98 at 09:17:03
From: Doctor Jerry
Subject: Re: trigonometric function

Hi Pusan High School students.

Your English looks good to me. The only comment I would make is that 
"trigonometry function" should be "trigonometric functions."

I think the third identity should always come from the first two.

I think that you know very well the standard book proof of the first 
two identities.

Since you said in VARIOUS WAYS, I think that you might be interested 
in the following.

It is true that e^{i*t} = cos(t) + i*sin(t). This is called Euler's 
identity. If it is okay to use this identity, then the first two 
identities can be derived easily.

Note:
  
   e^{i*x}*e^{i*y} = e^{i(x+y)}, by properties of exponents

and:

   [cos(x)+i*sin(x)]*[cos(y)+i*sin(y)] = cos(x+y)+i*sin(x+y)

Now, multiply the left side and separate it into real and imaginary 
parts, and you will have your first two identities.

I hope these comments were useful to you.

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