Trigonometric Recurrent Progression

Summary: §129 of Chapter 8. If the arcs $z,y+z,2y+z,3y+z,\dots$ form an arithmetic progression, then their sines (and their cosines) form a recurrent progression generated by the denominator $1-2nZ+(m^2+n^2)Z^2$, where $\sin y=m$ and $\cos y=n$. The recurrence is $\sin ((k+1)y+z)=2\cos y\cdot \sin (ky+z)-\sin ((k-1)y+z)$, and the same with cosine. This makes Chapter 4’s recurrent-series machinery applicable to the trigonometric tables.

Sources: chapter8 (§129)

Last updated: 2026-04-27

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Setup

Let $\sin z=p$, $\cos z=q$, $\sin y=m$, $\cos y=n$, with $p^2+q^2=1$ and $m^2+n^2=1$ from the Pythagorean identity. Repeated application of the §128 sum/difference formulas gives explicit polynomial expressions for $\sin (ky+z)$ and $\cos (ky+z)$ in $m,n,p,q$:

$k$	$\sin (ky+z)$	$\cos (ky+z)$
0	$p$	$q$
1	$mq+np$	$nq-mp$
2	$2mnq+(n^2-m^2)p$	$(n^2-m^2)q-2mnp$
3	$(3m{n}^2-m^3)q+(n^3-3m^{2}n)p$	$(n^3-3m^{2}n)q-(3m{n}^2-m^3)p$

(source: chapter8, §129). The sine and cosine columns are the imaginary and real parts of $(n+im{)}^k(q+ip)=e^{iky}(\cos z+i\sin z)$ — visible in retrospect once eulers-formula is established at §138, but Euler does not say so here.

The recurrence and its denominator

Reading down the columns: each entry equals $2n$ times the previous minus $(m^2+n^2)=1$ times the one before. Concretely,

$\sin ((k+1)y+z)=2n\cdot \sin (ky+z)-(m^2+n^2)\sin ((k-1)y+z),$

$\cos ((k+1)y+z)=2n\cdot \cos (ky+z)-(m^2+n^2)\cos ((k-1)y+z).$

Since $m^2+n^2=1$, the recurrence simplifies to

$\sin ((k+1)y+z)=2\cos y\cdot \sin (ky+z)-\sin ((k-1)y+z),$

with the identical recurrence for cosine. Euler writes it out for $k=1,2,3$ (source: chapter8, §129):

$\sin (2y+z)=2\cos y\cdot \sin (y+z)-\sin z,$

$\sin (3y+z)=2\cos y\cdot \sin (2y+z)-\sin (y+z),$

$\sin (4y+z)=2\cos y\cdot \sin (3y+z)-\sin (2y+z),$

and similarly for cosine. He emphasizes the practical advantage: “when the arcs form an arithmetic progression, then as many of the sines and cosines as may be desired can be expressed with little trouble.”

Reading as a recurrent series

In the language of Chapter 4, a sequence $a_0,a_1,a_2,\dots$ is recurrent with denominator $1+\alpha Z+\beta Z^2$ if $a_{k+1}=-\alpha a_k-\beta a_{k-1}$. Here $\alpha =-2n=-2\cos y$ and $\beta =m^2+n^2=1$, so the generating denominator is

$1-2\cos y\cdot Z+Z^2,$

or in Euler’s notation, $1-2nZ+(m^2+n^2)Z^2$. The roots of this quadratic are $Z=e^{\pm iy}=\cos y\pm i\sin y$ — the same complex numbers that drive De Moivre’s formula. The recurrent-progression description and the De Moivre description are two readings of the same underlying complex multiplication.

Why the recurrent-progression view matters

The §129 observation is more than a curiosity:

1. It gives a fast way to extend a sine table. Knowing $\sin z$ and $\sin (y+z)$ for a small arc $y$, every $\sin (ky+z)$ follows by two multiplications and one subtraction per step — no need to invoke addition formulas afresh. This is the algorithm used in many 18th-century practical computations.
2. It shows the trig recurrent progression has the same second-order structure as higher-order arithmetic progressions of Chapter 4 but with complex (rather than real, repeated) roots of the denominator. Real roots produce polynomial growth; complex unit-modulus roots produce bounded oscillation — the characteristic behavior of $\sin$ and $\cos$.
3. It is the bridge from Chapter 4’s algebraic recurrent series to the analytic series of §134. Once the recurrence is in hand, the limit $y\to 0$ converts difference equations into the differential structure that underlies the power series for $\sin v$ and $\cos v$.

Application in Chapter 14

Chapter 14 applies the same recurrence from a different angle: §234 uses the scale of relation $2y,-1$ (with $y=\cos z$ fixed and $n$ varying) to build the table of $\sin nz$, $\cos nz$ polynomials. Chapter 14 §258 then closes the circle by applying §231’s sum formula to compute ${\sum }_{k=0}^{\infty }\sin (a+kb)$ in closed form. See sum-of-trig-in-ap and multiple-angle-polynomials.

Related pages

• recurrent-series
• higher-order-arithmetic-progressions
• trigonometric-addition-formulas
• sine-and-cosine
• de-moivre-formula
• multiple-angle-polynomials
• sum-of-trig-in-ap
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle
• chapter-4-on-the-development-of-functions-in-infinite-series
• chapter-14-on-the-multiplication-and-division-of-angles