Maths Research Paper - Taylor Series

TRIGONOMETRY EXPLORATION by Willy Wibamanto
1. The graphs y= and y= intersect at the origin.
2. The graphs intersect at the origin.

3. As the degree of the polynomial increases, the graphs are approaching y=sin (x).

4. As the degree of the polynomial increases, the graphs are moving away from y=cos (x).

5a.
When y = sin (1), y = 0.841. Using the Taylor series with two terms, y = 0.830.
When y = sin (5), y = -0.958. Using the Taylor series with two terms, y = - 15.8.
When y = cos (1), y = 0.540. Using the Taylor series with two terms, y= 0.500.
When y = cos (5), y = 0.284. Using the Taylor series with two terms, y = - 11.5.

By using the formula, Percentage Error =
Percentage Error for Taylor series with two terms = = 1377.18 % ≈ 1380 % (3sf)
5b.
When y = sin (1), y = 0.841. Using the Taylor series with three terms, y = 0.842.
When y = sin (5), y = -0.958. Using the Taylor series with three terms, y = 10.2.
When y = cos (1), y = 0.540. Using the Taylor series with three terms, y = 0.542.
When y = cos (5), y = 0.284. Using the Taylor series with three terms, y = 14.5.

Percentage Error for Taylor series with two terms= = 1492.80% ≈ 1490% (3sf)
5c.
When y= sin (1), y = 0.841. Using the Taylor series with four terms, y= 0.841.
When y = sin (5), y = -0.958. Using the Taylor series with four terms, y= -5.29.
When y = cos (1), y = 0.540. Using the Taylor series with four terms, y = 0.540.
When y = cos (5), y = 0.284. Using the Taylor series with four terms, y = - 7.15.

Percentage Error for Taylor series with four terms= = 717.25% ≈ 717% (3sf)

OTHER STANDARD TAYLOR SERIES EXPANSIONS FOR DIFFERENT FUNCTIONS
1. Logarithmic Functions 2. Hyperbolic Functions
NON STANDARD TAYLOR SERIES EXPANSIONS
It is easy to take derivatives of Taylor series: Just take the