Expert: Josh - 2/25/2004

Question
I understand that if the angle (theta)is very small sin(theta)can be written as (theta) and cos(theta) can be written as 1. Is there any trignometric relations/derivations? Kindly help me out.

Answer
Hello,

Since these are analytic functions, if you consider their Taylor series expansion, it should be quite obvious.

I won't go into how you obtain the Taylor series expansion. You really need to understand the concept of convergence and Cauchy's theorem before you can appreciate the results and understand where the idea comes from. There is usually a course on complex analysis at university level.

Anyway, interesting to see that the Taylor series expansion for sin(x) is sin(x)=x-(x^3)/3!+(x^5)/5!-(x^7)/7!+...
This works for all real x values in the interval [0,2*pi).
If you then take the limit as x->0, you can obtain a good first-order approximation by truncating the series, and retaining the first term. That's where it comes from, but is not very intuitive until you have done some complex analysis.

P.S. cos(x)=1-(x^2)/2!+(x^4)/4!-(x^6)/6!+...
where factorial n!= n*(n-1)*(n-2)*...*2*1, eg., 4!=4*3*2*1.

CHeers.