Approximations of Sine
Description
This applet shows the Sine function and four approximations for the Sine function.
An early approximation for sine was from Bhaskara I is
. This function is fairly accurate for angles between .
Another approximation is a power series (MacLaurin or Taylor Series) which for the Sine function is
. The number of terms can be increased to obtain any desired accuracy.
The third approximation is a product series that was used in the proof of .
Since the Sine function is zero at
it can be approximated as . This is the Root Products Approximation.
A fourth method is Polynomial Interpolation that go through a set of points. Two points define a line. For more points the polynomial can be defined as using Lagrange Polynomials. Note, the value of the product is zero at all points and one when . Here the points are marked with small squares and are at well known Sine angles, where sine is . This is labeled Lebesgue.
Compare the different approximations as extra terms are added.
How many terms are required for the MacLaurin Series to achieve grater accuracy than the other approximations?
Where do the different approximations provide more accurate values?