Taylor Series: Common & Expansion Formula, and Examples

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.

Common Taylor Series

Taylor series formula

Common Taylor Series

It is also can be written as

Common Taylor Series 2

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

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 Expansion

Taylor Series Examples

  1. 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.

Taylor Series Example 1a

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.

Taylor Series Example 1b

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.

Taylor Series Example 2

So, the value of sin 3°» 0.0523.

Learn More


Arithmetic Sequences and Series Sums

Infinite Series

Geometric Sequences and Sums

Fibonacci Sequence