---
title: A Lightweight Derivation of CORDIC for Trigonometric Functions
author: Heath Davison Lunt
date: June 2026
---

Let $$\begin{aligned}
M=
\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{pmatrix}
\end{aligned}$$ It follows, by factoring out a $\cos\theta$ from $M$,
$$\begin{aligned}
M=
\cos\theta
\begin{pmatrix}
1 & -\tan\theta \\
\tan\theta & 1
\end{pmatrix}.
\end{aligned}$$

Set $$\begin{aligned}
\theta_0 = 0,
\quad
\theta_{n+1}
=
\sum_{k=0}^{n}
a_k \tan^{-1}\left(2^{-k}\right),
\end{aligned}$$ With $$\begin{aligned}
a_k
=
\begin{cases}
+1 & \text{if } \theta_k < \theta,\\
-1 & \text{if } \theta_k > \theta.
\end{cases}
\end{aligned}$$ That is to say, write $\theta$ as an infinite summation
of arctangents of the reciprocals of powers of two using the algorithm
seen in the definition of $a_{k}$, which can be surmised as follows:

> If the current angle approximation *overshoots* the target, subtract,
> otherwise, add.

Note that, for $$\begin{aligned}
\tan\alpha = 2^{-k},
\end{aligned}$$ we have $$\begin{aligned}
\cos\alpha
=
\frac{1}{\sqrt{1+2^{-2k}}}.
\end{aligned}$$

Hence, $$\begin{aligned}
M
=
\prod_{k=0}^{\infty}
\frac{1}{\sqrt{1+2^{-2k}}}
\begin{pmatrix}
1 & -a_{k}2^{-k} \\
a_{k}2^{-k} & 1
\end{pmatrix}.
\end{aligned}$$

We use the fact that $a_{k} \in \{-1 ,1\}$ as to move it within the
value of the tangent.

Splitting the product, $$\begin{aligned}
M
=
\left(
\prod_{k=0}^{\infty}
\frac{1}{\sqrt{1+2^{-2k}}}
\right)
\left(
\prod_{k=0}^{\infty}
\begin{pmatrix}
1 & -a_{k}2^{-k} \\
a_{k}2^{-k} & 1
\end{pmatrix}
\right).
\end{aligned}$$

We see that our left product is independent of $\theta$, and can be
treated as a constant. $$\begin{aligned}
\prod_{k=0}^{\infty}
\frac{1}{\sqrt{1+2^{-2k}}}
\approx
0.607252935\ldots
\end{aligned}$$ The decimal expansion of this constant is sequence
[A273413](https://oeis.org/A273413) in the OEIS

We can then write

$$\begin{aligned}
M
=
\left(0.607252935\ldots\right)
\prod_{k=0}^{\infty}
\begin{pmatrix}
1 & -a_{k}2^{-k} \\
a_{k}2^{-k} & 1
\end{pmatrix}.
\end{aligned}$$

Expanding this infinite product into an iterative formula, we get

$$\begin{aligned}
\begin{aligned}
x_{k+1} &= x_k - a_k 2^{-k} y_k,\\
y_{k+1} &= y_k + a_k 2^{-k} x_k,
\end{aligned}
\qquad k\in\mathbb{N}\cup\{0\}
\end{aligned}$$

$$\begin{aligned}
\lim_{n\to\infty}
\begin{pmatrix}
0.60725\ldots\cdot x_n\\
0.60725\ldots\cdot y_n
\end{pmatrix}
=
M
\begin{pmatrix}
x\\
y
\end{pmatrix}.
\end{aligned}$$