Multiple-Angle Polynomials

Summary: §234–§238, §243. Euler derives explicit polynomial formulas expressing $\sin nz$ and $\cos nz$ as polynomials in $\sin z$ and $\cos z$, using the recurrence $\sin ((k+1)z)=2\cos z\cdot \sin (kz)-\sin ((k-1)z)$ (scale of relation $2y,-1$). For odd $n$, $\sin nz$ reduces to a pure polynomial in $x=\sin z$; for even $n$, there is a residual factor $\sqrt{1-x^2}$.

Sources: chapter14 (§234–§238, §243)

Last updated: 2026-05-10

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Setup (§234)

Let $z$ be any arc on the unit circle. Write $x=\sin z$, $y=\cos z$, $t=\tan z$, so that $x^2+y^2=1$ and $t=x/y$. Both the sine and cosine series of $z,2z,3z,\dots$ are recurrent with scale of relation $2y,-1$ — the same structure as §129 but applied to the sequence indexed by the multiplier $n$ rather than an additive offset.

Sine table (§234)

$n$	$\sin nz$
0	$0$
1	$x$
2	$2xy$
3	$4x{y}^2-x$
4	$8x{y}^3-4xy$
5	$16x{y}^4-12x{y}^2+x$
6	$32x{y}^5-32x{y}^3+6xy$
7	$64x{y}^6-80x{y}^4+24x{y}^2-x$
8	$128x{y}^7-192x{y}^5+80x{y}^3-8xy$

The general formula (source: chapter14, §234):

$\sin nz=x\left[2^{n-1}{y}^{n-1}-\frac{(n-2)}{1}{2}^{n-3}{y}^{n-3}+\frac{(n-3)(n-4)}{1\cdot 2}{2}^{n-5}{y}^{n-5}-\cdots \right]$

Every term carries the factor $x=\sin z$, reflecting the functional identity $\sin nz=0$ when $z=0$.

Odd $n$: pure polynomial in $\sin z$ (§236)

For odd $n$, substituting $y=\sqrt{1-x^2}$ eliminates $y$ entirely, because $x$ factors from every term and the remaining polynomial in $y^2$ becomes a polynomial in $x^2$. The resulting formula is

$\sin nz=nx-\frac{n(n^2-1)}{1\cdot 2\cdot 3}{x}^3+\frac{n(n^2-1)(n^2-9)}{1\cdot 2\cdot 3\cdot 4\cdot 5}{x}^5-\frac{n(n^2-1)(n^2-9)(n^2-25)}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}{x}^7+\cdots$

(source: chapter14, §236). The numerator factors in the general term are $n^2-(2k-1{)}^2$ for $k=1,2,3,\dots$. Sample cases:

• $n=3$: $\sin 3z=3x-4x^3$
• $n=5$: $\sin 5z=5x-20x^3+16x^5$
• $n=7$: $\sin 7z=7x-56x^3+112x^5-64x^7$

These are precisely the Chebyshev polynomials of the second kind (up to normalization); Euler does not use this name.

Even $n$: residual radical (§238)

For even $n$, each term in the sine table has a factor $y$, so

$\sin nz=\left[nx-\frac{n(n^2-4)}{1\cdot 2\cdot 3}{x}^3+\frac{n(n^2-4)(n^2-16)}{1\cdot 2\cdot 3\cdot 4\cdot 5}{x}^5-\cdots \right]\sqrt{1-x^2}$

(source: chapter14, §238). To obtain a pure polynomial equation, Euler squares both sides and rearranges:

$\frac{(\sin nz{)}^2}{1-x^2}=n^2{x}^2-\cdots$

giving a degree-$2n$ polynomial equation in $x$ whose $2n$ roots are both positive and negative (source: chapter14, §239).

Sample cases:

• $n=2$: $\sin 2z=2x\sqrt{1-x^2}$
• $n=4$: $\sin 4z=(4x-8x^3)\sqrt{1-x^2}$
• $n=6$: $\sin 6z=(6x-32x^3+32x^5)\sqrt{1-x^2}$

Cosine table (§243)

Using $\cos z=y$ as the base variable:

$n$	$\cos nz$
0	$1$
1	$y$
2	$2y^2-1$
3	$4y^3-3y$
4	$8y^4-8y^2+1$
5	$16y^5-20y^3+5y$
6	$32y^6-48y^4+18y^2-1$
7	$64y^7-112y^5+56y^3-7y$

The general formula (source: chapter14, §243):

$\cos nz=2^{n-1}{y}^n-\frac{n}{1}{2}^{n-3}{y}^{n-2}+\frac{n(n-3)}{1\cdot 2}{2}^{n-5}{y}^{n-4}-\frac{n(n-4)(n-5)}{1\cdot 2\cdot 3}{2}^{n-7}{y}^{n-6}-\cdots$

Since $\cos nz=\cos (2\pi \pm nz)=\cos (4\pi \pm nz)=\cdots$, the $n$ roots of the cosine polynomial are $\cos z,\cos (\frac{2\pi }{n}\pm z),\cos (\frac{4\pi }{n}\pm z),\dots$

These are the Chebyshev polynomials of the first kind $T_n(y)$.

Connection to the recurrent series (§234)

The table is generated by the two-term recurrence

$\sin ((k+1)z)=2y\cdot \sin (kz)-\sin ((k-1)z)$

with initial values $\sin 0z=0$, $\sin 1z=x$. This is the same scale of relation $2y,-1$ identified in §129, but now viewed as a recurrence in $n$ rather than in an additive offset. The scale of the relation machinery from Chapter 13 therefore applies directly.

Why it matters

The multiple-angle polynomials are the algebraic heart of sine-cosine-factored-products and trig-values-as-roots: the roots of the odd-$n$ polynomial in $x$ are exactly the $n$ values of $\sin (z+2k\pi /n)$, and reading Vieta’s formulas off the polynomial coefficients gives the partial-fraction and product identities that dominate the rest of Chapter 14.

Related pages

• sine-cosine-factored-products
• trig-values-as-roots
• de-moivre-formula
• trigonometric-recurrent-progression
• scale-of-the-relation
• sine-and-cosine
• chapter-14-on-the-multiplication-and-division-of-angles
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle