# Exact values of trig functions

• Dec 26th 2012, 06:10 PM
kinhew93
Exact values of trig functions
Just a quick question - How are the values of the basic trig functions found? Other than the few that can be found exactly from particular triangles are the rest just found by drawing/measuring as accurately possible or can we find the exact value of any trig ratio?

I hope my question makes sense...
• Dec 26th 2012, 07:12 PM
zhandele
Re: Exact values of trig functions
You can express the sine, cosine or tangent of any number as the sum of an infinite series. Generally, the sum will be an irrational number. In real-life applications, we approximate the values to whatever number of digits are needed. There is a method that will tell us how many terms in the infinite series we must evaluate in order to reach a particular level of accuracy. You will learn about this when you take calculus. In the meanwhile, I wouldn't worry about it. Your calculator or CAS will evaluate any trig functions you need evaluated.
• Dec 26th 2012, 07:22 PM
Soroban
Re: Exact values of trig functions
Hello, kinhew93!

I had the same question years ago.
Way back, when we had 4-place trig tables to work with.

A few years later, I found out how in a Calculus course.

There are infinite series for the basic trig functions.

For example: . $\begin{Bmatrix}\sin x &=& x - \dfrac{x^3}{3!}+\dfrac{x^5}{5!} -\dfrac{x^7}{7!} + \cdots \\ \\[-3mm] \cos x &=& 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots \end{Bmatrix}$ . where $x$ is in radians.

Armed with these formulas (and a calculator),
. . we can crank out the sine and cosine values for any angle
. . to any number of decimal places.