More Approximations for Trigonometric Functions with the Binomial Series


Posted on August 31, 2015


In this post, we will create a new set of approximations for sine and cosine that utilize the binomial series. Before you go any further, you may want to read this previous post where we walk through the binomial series and comparisons to the Maclaurin series.

Derivation

We are going to exploit Euler's formula with a twist. To refresh your memory, here's Euler's Formula. \begin{equation} \cos(\theta)+i\sin(\theta)=e^{i\theta} \end{equation} Now let's employ one of the older tricks in the book, adding and subtracting one from the same expression: \begin{equation} \cos(\theta)+i\sin(\theta)=\left[(e-1)+1\right]^{i\theta}. \end{equation} Now we have a binomial raised to a power. Once again, we will use known values and the binomial series to generate approximations for sine and cosine.

Fine Tuning For Approximation

In its current form this would almost be enough for decent approximations. However, closely tied to the accuracy of an approximation is the convergence rate of the series. To improve convergence, we want to make one of the terms small relative to the other term. Is there anything we can do here?

As a matter of fact, yes. We will employ a second manipulation - multiplying and dividing by the same number (inside and outside of the parentheses). Consider the updated equation \begin{equation} \cos(\theta)+i\sin(\theta)=a^i\cdot\left[\left(\displaystyle\frac{e}{a^{1/\theta}}-1\right)+1\right]^{i\theta}. \end{equation} If you need to, convince yourself that this is still valid. Now, to build a valid approximation, we need $\displaystyle\frac{e}{a^{1/\theta}}-1$ to be small for a range of values. In our previous post, we determined that you need a range of $\pi/4$ for sine and cosine, which could be extrapolated out to encompass all values.

Ideally we would just pick the range $[0,\pi/4]$, with which a value of $a=e^{\pi/8}$ would be a good place to center the approximation. However, If $\theta=0$, the expression \begin{equation} {(e^{\pi/8})^{1/0}}-1 \end{equation} is not defined (although the limit appears to exist).

On the other side of the range, at $\theta=\pi/4$, the expression becomes \begin{equation} \displaystyle\frac{e}{(e^{\pi/8})^{4/\pi}}-1=e^{1/2}-1\approx.64 \end{equation} which is enough for convergence, but maybe not optimal.

On the other hand, if we made the range $[2\pi,2\pi+\pi/4]$ with $a=2\pi+\pi/8$, then our bounding values are roughly $\approx\pm.06$ which will give us much better convergence. Due to the nature of this equation, rapidly increasing our range - for example, the range $[100\pi,100\pi+\pi/4]$ - has rapidly diminishing gains, and improved convergence will be offset by roundoff error elsewhere.

Implementing Approximations

An example of this code is (written for MATLAB) is

iterations=14;
start=2*pi;
range=pi/4;
increment=.001;

theta=start:increment:(start+range);

a=exp(start+range/2);
z=1i.*theta;
w=exp(1)./a.^(1./theta)-1;

approx=zeros(1,length(theta));

for i=iterations:-1:1
    approx=1+approx.*(w.*(z-i+1)/i);
end
         
approx=approx*a.^(1i);

You can plot the error with the following code

error=real(approx)-cos(theta);
figure; 
h=semilogy(theta,abs(error));
outfilename=sprintf('cos_error_%d_iter',iterations);
saveas(h, outfilename, 'png');

error=imag(approx)-sin(theta);
figure; 
h=semilogy(theta,abs(error));
outfilename=sprintf('sin_error_%d_iter',iterations);
saveas(h, outfilename, 'png');

Error Plots

See below for the error for approximations for cosine with 4, 8, 12, and 14 terms. With 14 terms, we are more or less bounded by machine precision (~15 significant figures)

Error plot for $\cos(\theta)$ approximated with 4 terms on the range $\theta=[2\pi,2\pi+\pi/4]$
Error plot for $\cos(\theta)$ approximated with 8 terms on the range $\theta=[2\pi,2\pi+\pi/4]$
Error plot for $\cos(\theta)$ approximated with 12 terms on the range $\theta=[2\pi,2\pi+\pi/4]$
Error plot for $\cos(\theta)$ approximated with 14 terms on the range $\theta=[2\pi,2\pi+\pi/4]$
Error plot for $\sin(\theta)$ approximated with 14 terms on the range $\theta=[2\pi,2\pi+\pi/4]$

Once again we have generated some exciting looking, but nonetheless useless approximations. We were able to use binomial theorem and some trickery to generate approximations with gerat accuracy, but once again they are nowhere near as speedy to calculate as the Maclaurin series.