sine-cosine-factored-products

Sine and Cosine Factored Products

Summary: §237, §240–§242, §245. The product of the $n$ roots of the multiple-angle polynomial for $\sin nz$ (or $\cos nz$) yields a factorization of $\sin nz$ (or $\cos nz$) as $2^{n-1}$ times a product of $n$ sines (or cosines) of shifted angles. A single unified formula covers both odd and even $n$.

Sources: chapter14 (§237, §240–§242, §245)

Last updated: 2026-05-10

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Odd $n$: product formula for $\sin nz$ (§237)

From §236–§237, the $n$ roots of the odd-$n$ polynomial in $x=\sin z$ are $\sin z,\sin (\frac{2\pi }{n}+z),\sin (\frac{4\pi }{n}+z),\dots$. The leading coefficient of the polynomial is $\pm 2^{n-1}$ (from the table in §236). The constant term divided by the leading coefficient is $\sin nz/(\pm 2^{n-1})$. Writing out the product of all roots and equating:

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

(source: chapter14, §237). Because $\sin v=\sin (\pi -v)$, the root $\sin (\frac{(n-k)\pi }{n}+z)$ equals $\sin (\frac{k\pi }{n}-z)$ for appropriate $k$, so the product telescopes to $n$ factors total.

Example $n=3$ (§237):

$\sin 3z=-4\sin z\cdot \sin (\frac{2\pi }{3}+z)\cdot \sin (\frac{4\pi }{3}+z)=4\sin z\sin (\frac{\pi }{3}-z)\sin (\frac{\pi }{3}+z)$

Example $n=5$ (§237):

$\sin 5z=16\sin z\cdot \sin (\frac{\pi }{5}-z)\sin (\frac{\pi }{5}+z)\sin (\frac{2\pi }{5}-z)\sin (\frac{2\pi }{5}+z)$

Even $n$: product formula for $\sin nz$ (§239–§240)

For even $n$, squaring removes the $\sqrt{1-x^2}$ factor. The $2n$ roots of the resulting polynomial come in positive/negative pairs. From the product formula for the squared version, taking the square root and identifying signs:

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

(source: chapter14, §239–§240). Example $n=2$:

$\sin 2z=2\sin z\cdot \sin (\frac{\pi }{2}-z)=2\sin z\cos z$

Example $n=4$:

$\sin 4z=8\sin z\cdot \sin (\frac{\pi }{4}-z)\sin (\frac{\pi }{4}+z)\sin (\frac{\pi }{2}-z)$

Unified formula (§241)

Euler observes that both cases are captured by the single expression (§241):

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

with exactly $n$ factors, the last factor being $\sin (\frac{\pi }{2}-z)=\cos z$ when $n$ is even, and $\sin z$ itself paired with the outermost product terms when $n$ is odd. The full table from §241:

$n$	$\sin nz$
1	$\sin z$
2	$2\sin z\cos z$
3	$4\sin z\cdot \sin (\frac{\pi }{3}-z)\sin (\frac{\pi }{3}+z)$
4	$8\sin z\cdot \sin (\frac{\pi }{4}-z)\sin (\frac{\pi }{4}+z)\sin (\frac{2\pi }{4}-z)$
5	$16\sin z\cdot \sin (\frac{\pi }{5}-z)\sin (\frac{\pi }{5}+z)\sin (\frac{2\pi }{5}-z)\sin (\frac{2\pi }{5}+z)$
6	$32\sin z\cdot \sin (\frac{\pi }{6}-z)\sin (\frac{\pi }{6}+z)\sin (\frac{2\pi }{6}-z)\sin (\frac{2\pi }{6}+z)\sin (\frac{3\pi }{6}-z)$

Cosine products (§242, §245)

Using the identity $\cos nz=\sin 2nz/(2\sin nz)$ (§242), the cosine products follow from the sine products at $n$ and $2n$. Euler lists:

$\cos nz=2^{n-1}\sin (\frac{\pi }{2n}-z)\sin (\frac{\pi }{2n}+z)\sin (\frac{3\pi }{2n}-z)\sin (\frac{3\pi }{2n}+z)\cdots$

where there are $n$ factors (source: chapter14, §242).

In §245 a cosine-only version is derived. Using $\cos v=-\cos (\pi -v)$:

$\cos nz=2^{n-1}\cos (\frac{n-1}{n}\pi +z)\cos (\frac{n-1}{n}\pi -z)\cos (\frac{n-3}{n}\pi +z)\cdots$

Sample cases from §245:

• $\cos z=\cos z$
• $\cos 2z=2\cos (\frac{\pi }{4}+z)\cos (\frac{\pi }{4}-z)$
• $\cos 3z=4\cos (\frac{2\pi }{6}+z)\cos (\frac{2\pi }{6}-z)\cos z$
• $\cos 4z=8\cos (\frac{3\pi }{8}+z)\cos (\frac{3\pi }{8}-z)\cos (\frac{\pi }{8}+z)\cos (\frac{\pi }{8}-z)$

Relation to the sine-infinite-product

The §241 formula at $z$ fixed and $n\to \infty$ recovers the §158 infinite product $\sin z=z{\prod }_{k=1}^{\infty }(1-z^2/k^2{\pi }^2)$: each finite factor $\sin (k\pi /n\pm z)$ approximates $(k\pi /n\pm z)$ for large $n$, and the $2^{n-1}$ prefactor accounts for the $z$ in the numerator. Chapter 14’s finite products are thus a finite-$n$ refinement of Chapter 9’s infinite product.

Sum of sines of equally-spaced angles (§237)

The penultimate Vieta formula (sum of all roots, §237) gives

$0=\sin z+\sin (\frac{2\pi }{n}+z)+\sin (\frac{4\pi }{n}+z)+\cdots$

(sum of $n$ equally-spaced sines is zero). This is the $n$-point discrete Fourier sum, which Euler derives here for the first time in this generality.

Related pages

• multiple-angle-polynomials
• trig-values-as-roots
• trig-multiple-angle-partial-fractions
• sine-infinite-product
• cosine-infinite-product
• factorization-of-an-plus-minus-zn
• chapter-14-on-the-multiplication-and-division-of-angles