Chapter 14 — On the Multiplication and Division of Angles

Summary: §234–§263. Euler systematically derives polynomial and product expressions for $\sin nz$, $\cos nz$, $\tan nz$ in terms of trig functions of $z$, identifies the roots of the resulting polynomial equations as trig values at equally-spaced angles, and reads off partial-fraction, sum, and product relations for all six trig functions at multiple angles. He then sums sines and cosines of arithmetic progressions (both infinite and finite), and closes by inverting the multiple-angle polynomials to express any power $(\sin z{)}^n$, $(\cos z{)}^n$ as a binomial-weighted linear combination of sines or cosines of multiple angles.

Sources: chapter14

Last updated: 2026-05-11

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Movement 1 — Multiple-angle polynomials (§234–§238, §243)

Starting from the recurrence $\sin ((k+1)z)=2\cos z\cdot \sin (kz)-\sin ((k-1)z)$ (scale of relation $2y,-1$ with $y=\cos z$), Euler builds the tables for $\sin nz$ and $\cos nz$ up to $n=8$. See multiple-angle-polynomials for the general formulas.

For odd $n$, substituting $y=\sqrt{1-x^2}$ (where $x=\sin z$) collapses $\sin nz$ to a pure polynomial in $x$:

$\sin nz=nx-\frac{n(n^2-1)}{1\cdot 2\cdot 3}{x}^3+\frac{n(n^2-1)(n^2-9)}{1\cdot 2\cdot 3\cdot 4\cdot 5}{x}^5-\cdots$

For even $n$, a factor $\sqrt{1-x^2}$ remains; squaring yields a polynomial equation.

For $\cos nz$ (§243), the base variable is $y=\cos z$ and the analogous polynomial is

$\cos nz=2^{n-1}{y}^n-\frac{n}{1}{2}^{n-3}{y}^{n-2}+\frac{n(n-3)}{1\cdot 2}{2}^{n-5}{y}^{n-4}-\cdots$

Movement 2 — Roots as trig values at equally-spaced angles (§235–§236, §239, §243–§244)

The equation $\sin s=\sin nz$ has $n$ solutions

$z=\frac{s}{n},\frac{\pi -s}{n},\frac{2\pi +s}{n},\frac{3\pi -s}{n},\dots$

These are the $n$ roots of the polynomial in $x$. From Vieta’s formulas Euler reads off: the sum of all roots is zero; the sum of their reciprocals is $n/\sin nz$; and the product gives $\sin nz$ as a product of shifted sines (§237). See trig-values-as-roots.

Movement 3 — Factored products for sin and cos (§237, §240–§242, §245)

The central product formula (odd and even $n$ unified in §241):

$\sin nz=2^{n-1}\sin z\cdot \sin (\frac{\pi }{n}-z)\sin (\frac{\pi }{n}+z)\sin (\frac{2\pi }{n}-z)\sin (\frac{2\pi }{n}+z)\cdots$

where the number of factors equals $n$. Using $\cos nz=\sin 2nz/(2\sin nz)$ (§242), matching cosine products are derived. See sine-cosine-factored-products.

Movement 4 — Partial-fraction sums for csc, sec, cot, tan (§237, §246–§256)

From the Vieta reciprocal-root sum, Euler derives:

$\frac{n}{\sin nz}=\frac{1}{\sin z}+\frac{1}{\sin (\frac{2\pi }{n}+z)}+\frac{1}{\sin (\frac{4\pi }{n}+z)}+\cdots$

and analogous expansions for $n\cot nz$, $n\sec nz$, $n\csc nz$ (§246–§248). For the tangent (§249–§256), De Moivre gives $\tan nz$ as a rational function of $t=\tan z$, from which $n\cot nz$ splits into $n$ cotangents. See trig-multiple-angle-partial-fractions.

Movement 5 — Sum of sines/cosines in arithmetic progression (§258–§260)

Since sines of equally-spaced angles form a recurrent series, the infinite sum $\sin a+\sin (a+b)+\sin (a+2b)+\cdots$ is the rational function

$s=\frac{\sin a-\sin (a-b)}{2-2\cos b}=\frac{\sin a-\sin (a-b)}{2(1-\cos b)}$

obtained by evaluating the generating function at $z=1$. The finite sum through $\sin (a+nb)$ is then obtained (§259–§260) by subtracting the corresponding tail, giving the standard

${\sum }_{k=0}^n\sin (a+kb)=\frac{\sin (a+\frac{1}{2}nb)\sin (\frac{1}{2}(n+1)b)}{\sin (\frac{1}{2}b)}$

and the analogous cosine formula. See sum-of-trig-in-ap.

Movement 6 — Powers of sin and cos as multiple-angle sums (§261–§263)

Inverting the chapter’s main thread: any power $(\sin z{)}^n$ is a binomial-weighted finite sum of $\sin kz$ (or $\cos kz$ for even $n$), and similarly for $(\cos z{)}^n$. Examples:

$4(\sin z{)}^3=3\sin z-\sin 3z,8(\cos z{)}^4=3+4\cos 2z+\cos 4z.$

The reduction is obtained from the four product-to-sum identities of §262. See powers-of-sine-and-cosine.

Related pages

• multiple-angle-polynomials
• sine-cosine-factored-products
• trig-values-as-roots
• trig-multiple-angle-partial-fractions
• sum-of-trig-in-ap
• powers-of-sine-and-cosine
• de-moivre-formula
• trigonometric-recurrent-progression
• cotangent-partial-fraction
• recurrent-series
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle
• chapter-13-on-recurrent-series