Derivation of the Power Series of Cosine

Date: 1/22/96 at 15:51:29
From: Anonymous
Subject: Taylor Polynomials

Would you please explain how the power series is derived from
the cosine function and any other function?  I am pretty confused 
about this and could use some explanation. Thank you.

Date: 6/21/96 at 11:56:20
From: Doctor Jerry
Subject: Re: Taylor Polynomials

I'm not at all certain about your question.  Perhaps I can say a 
few things and then wait for your response.

The cosine function, along with the other trig functions, inverse 
trig functions, logarithms, and exponentials, can't be calculated 
exactly except in special cases like cos(pi/2) or cos(pi/3).  
It's different from a polynomial like x^3-2x+1.  To evaluate this 
function, choose an x, cube it and substract two times it, and 
add 1.  

For cosine, there is no such finite procedure. There is, however, 
a sequence of polynomial approximations to cosine.  Here are the 
first several:

p1 = 1
p2 = 1 - x^2/2!
p3 = 1 - x^2/2! + x^4/4!
p4 = 1 - x^2/2! + x^4/4! - x^6/6!

You can try, on your calculator (set in radians), cos(0.1), first 
by pressing the COS key.  Then try p1, p2, p3, and p4, with x=0.1.

I did this and got 

1
0.995
0.9951
0.995004166667
0.995004165278

I found cos(0.1) = 0.995004165278.  You can see that p4 with 
x = 0.1 gives a very good approximation to cos(0.1).

The polynomials p1,p2,p3,p4,...are the partial sums of the power 
series for cos(x).  Partial sums just means the sum of the first 1 
terms, the sum of the first 2 terms, and so on.

The series for cos(x) is written as 

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

With an infinite number of terms, it is exact.  But to actually 
evaluate cos(x) for an x you must choose an x and then decide how 
many terms you need to add to get the accuracy you want.

There is a formula for figuring out power series called Taylor's 
formula.  It involves the derivatives of the function.

I hope I've answered some parts of your question.

-Doctor Jerry,  The Math Forum