Posted on April 4, 2014
In this post, we will introduce the incredibly useful binomial theorem and series, then use the series and De Moivre's theorem to derive a representation and approximations for trigonometric functions.
The binomial theorem is a way to expand a binomial raised to a power. For a natural number $n \in \mathbb{N}$ the binomial theorem is $$ (a+b)^n={n \choose 0} a^n b^0 + {n \choose 1} a^{n-1}b+{n \choose 2} a^{n-2}b^2…+{n \choose n} a^0 b^{n}.$$
A tidier closed form expression is $$ (a+b)^n=\sum_{k=0}^{n}{ {n \choose k} a^{n-k}b^{k} } .$$ Closely related to the binomial theorem (integer exponents) is the binomial series (for non-integer exponents).
For a real exponent $\alpha \in \mathbb{R}$ the binomial theorem becomes: $$ (a+b)^{\alpha}=\sum_{k=0}^{\infty}{ {\alpha \choose k} a^{\alpha-k}b^{k} } .$$ Without getting too in depth on convergence, it is important that $a<b$. The convergence of this series when $a=b$ depends on $\alpha$. There is more on convergence here.
Even if $\alpha$ isn't an integer, the coefficients ${ \alpha \choose k}$ are calculated similarly. Let’s look at two examples.
For integers: $${ 7 \choose 3}=\frac{7\cdot 6\cdot 5}{3\cdot 2\cdot 1}=35.$$ For non-integers: $${ 7.2 \choose 3}=\frac{7.2\cdot 6.2\cdot 5.2}{3\cdot 2\cdot 1}=38.688.$$ The generalized binomial has some useful applications like approximating roots, deriving multiple angle formulas, and approximating $e$.
The starting point for deriving these series is De Moivre's Theorem. The theorem is most commonly in used complex analysis, but it is probably best known as a tool for deriving multiple angle formulas. This theorem can be found in any complex analysis text, but we will state it here for clarity.
De Moivre's Theorem: For $n\in \mathbb{Z}$ and $\phi \in \mathbb{R}$
In multiple angle formula derivations, $n$ is generally a constant integer and $\phi$ is a variable. However, there are two significant differences in our approximations. We will leave $\phi$ constant and we will replace $n$ with a variable $\alpha \in \mathbb{R}$ that can vary.
In general, De Moivre's Theorem breaks down when generalized to $\alpha \in \mathbb{R}$. Any standard complex analysis book can provide examples of this. The proof is a little dry so we won’t show it here, but It can be shown that as long as we pick $\phi \in (-\pi,\pi]$, our generalization of De Moivre's Theorem holds.
Let $\phi \in (-\pi,\pi]$ with $\phi \neq 0$ and let $\theta=\alpha\phi$ for $\alpha \in \mathbb{R}$. Now we can take this special case of De Moivre's Theorem and apply the generalized binomial theorem, yielding: \begin{eqnarray*} \cos{\theta}+i\sin{\theta}&=&\cos{\alpha\phi}+i\sin{\alpha\phi}\\ &=&(\cos{\phi}+i\sin{\phi})^\alpha \\ &=&\displaystyle\sum_{k=0}^\infty{i^k{\alpha\choose k}\cos^{\alpha-k}{\phi}\sin^{k}{\phi} } \\ &=&\cos^{\alpha}{\phi}\displaystyle\sum_{k=0}^\infty{i^k{\alpha\choose k}\tan^{k}{\phi} }\\ &=&\cos^{\alpha}{\phi}\displaystyle\sum_{k=0}^\infty{(-1)^k\left[{\alpha\choose 2k}\tan^{2k}{\phi}+ i{\alpha \choose 2k+1}\tan^{2k}{\phi}\right]}. \end{eqnarray*} Now we equate the real and imaginary parts: \begin{eqnarray*} \cos{\theta}&=&\cos^{\alpha}{\phi}\displaystyle\sum_{k=0}^\infty{(-1)^k{\alpha\choose 2k}\tan^{2k}{\phi} } \\ \sin{\theta}&=&\cos^{\alpha}{\phi}\displaystyle\sum_{k=0}^\infty{(-1)^k{\alpha \choose 2k+1}\tan^{2k+1}{\phi} }. \end{eqnarray*} We can also create an alternative form by letting $x=\tan\phi \rightarrow \cos\phi=1/\sqrt{1+x^2}$: \begin{eqnarray*} \cos{\theta}&=&\left(1+x^2\right)^{-\alpha/2}\displaystyle\sum_{k=0}^\infty{(-1)^k{\alpha\choose 2k}x^{2k} } \\ \sin{\theta}&=&\left(1+x^2\right)^{-\alpha/2}\displaystyle\sum_{k=0}^\infty{(-1)^k{\alpha \choose 2k+1}x^{2k+1} }. \end{eqnarray*} While we will not go in depth here, it is important to study the convergence of these series. As a general rule, the series will converge for $\phi$ such that $\cos{\phi}>\sin{\phi}$ (this can be verified with the ratio test). It can also be shown with the alternating series test that if $\cos{\phi}=\sin{\phi}$, then the series will converge for $\theta\geq-\pi/4$. Both series converge for the values of $\theta$ and $\phi$ chosen later in this paper.
To implement these series as approximations, they must be truncated. The error associated with this truncation is heavily dependent on both $n$, the number of terms in the truncated series, and the value of $\phi$. The number of terms will be predetermined for consistency and computational efficiency. Realistically, values around $n=8$ are a good starting place because the standard approximation for trigonometric functions in MATLAB utilizes a truncated and slightly modified Maclaurin series with $8$ terms.
For the approximations to be repeatable without re-evaluating $\tan{\phi}$ and $\cos{\phi}$, the value of $\phi$ must be constant. With $\phi$ and $n$ constant, $\alpha$ will be recomputed for each $\theta$. Sensitivity studies on $\phi$ showed that approximations with $\tan{\phi}=.1$ are exceptionally accurate for a small number of terms, so that value will be used for approximations throughout the rest of the paper. Let’s try an example.
Example: Approximate $\cos{\pi/3}$.
Solution:Let $\phi=\tan^{-1}({.1})\approx.0997$. Then we compute $\alpha = \frac{\pi/3}{.0997} = 10.5.$ We will use $n=8$ terms in our approximation. We have \begin{eqnarray*} \cos{\pi/3} &\approx& \cos^{10.5}({.0997}) \displaystyle\sum_{k=0}^7{(-1)^k{10.5 \choose 2k}\tan^{2k}({.0997})} \\ &=&\cos^{10.5}({.0997}) \left[ 1-\frac{10.5(10.5-1)}{2!}.1^2+-... \ +-\frac{10.5(10.5-1)...(10.5-13)}{14!}.1^{14} \right] \\ &=&0.49999999999999944.\\ \end{eqnarray*} These approximations cannot be easily evaluated by hand, but can be carried out with a computer quickly and with exceptional accuracy. The outline and basic code for the cosine approximation are below.
An example of this code is (written for MATLAB) is
[approx, error] = function calculate_theta(theta)
tan_phi=.1;
phi=0.099668652491162;
cos_phi=0.995037190209989;
alpha=theta/phi;
coeff=1;
n=8;
sum=0;
for i=1:n
sum=sum+coeff;
coeff=-tan_phi^2*coeff*(alpha-2*i+1)*(alpha-2*i+2)/(2*i)/(2*i-1);
end
approx=sum*cos_phi^alpha;
error=approx-cos(theta);
Recall, $\phi$ is predetermined, so $\tan{\phi}$ and $\cos{\phi}$ are constants. In this code, they are represented with tan_phi and cos_phi respectively. Also, theta represents $\theta$, alpha represents $\alpha$, n represents the number of terms in the truncated series, coeff is each term in the series, and sum is utilized in the loop to add up the terms.
Before we get too ahead of ourselves, we need to consider how this stacks up against existing approximations for trigonometric functions. The simplest is the Maclaurin Series, with example source code shown here. While our approximations are a novel new look, they take much longer to compute so they are not a viable replacement for any existing approximations.