Taylor series is one of topic in numerical method. It describes the sum of infinite terms of any functions. If there are more and more terms, it will give the high accuracy.
It was introduced by English mathematicians, Brook Taylor. One of Taylor series application is scientific calculator operation.
Sometimes when we need to determine value of any angles in trigonometry and we need scientific calculator to make it easy. Scientific calculator use Taylor series to determine trigonometry values.
Generally, Taylor series is used to determine approximately of the function’s value by using polynomial’s concept.
Table of Contents
Common Taylor Series
Taylor series formula
It is also can be written as
Based on the common Taylor series formula and concept, there are some Taylor series formula that can be used to calculate trigonometry and others.
Taylor Series Expansion
Taylor series can be expanded to be another series. It was introduced by Scotland mathematicians Colin Maclaurin. So, it is called Maclaurin series.
Maclaurin series is the Taylor series that has specific case. It is Taylor series that has c = 0. The formula became:
Taylor Series Examples
- Describe sin (x) and cos (x) to be Maclaurin series.
a. Sin (x)
First, define derivative of sin (x)
F(x) = sin (x)
F’(x) = cos (x)
F’’(x) = – sin (x)
F’’’(x) = – cos (x)
F’’’’(x) = sin (x)
…..
The Maclaurin series is:
*remember that Maclaurin series is Taylor series with c = 0.
b. Cos (x)
First, define derivative of cos (x)
F(x) = cos (x)
F’(x) = – sin (x)
F’’(x) = – cos (x)
F’’’(x) = sin (x)
F’’’’(x) = cos (x)
…..
The Maclaurin series is:
*remember that Maclaurin series is Taylor series with c = 0.
2. Using Taylor series, calculate the value of sin 3°!
- Convert to be radian value
180° = p rad
1° = ( p / 180°) rad
= 3.1415926535897932384626433832795/180°
= 0.01745329251994329576923690768489
3° = 3 x 0.0174532925199433 (just take some digits)
= 0.05236 (make it simpler by taking 6 digits)
- Substitute to Taylor series
Numbers of terms on Taylor series depend on each person. It can use 2/3/4/5 terms. If it uses more and more terms to calculate it will give higher accuracy. In this example, we use three terms.
So, the value of sin 3°» 0.0523.