Trig Values as Roots
Summary: §235–§236, §239, §243–§244, §250–§256. The multiple-angle polynomial equation has exactly distinct roots, which are the sines of equally-spaced arcs . Vieta’s formulas on these roots produce the partial-fraction sums and products of trig functions that occupy §237–§257.
Sources: chapter14 (§235–§256)
Last updated: 2026-05-10
The roots of (§235)
If we set , then , so
There are exactly distinct values of corresponding to these arcs, namely
(source: chapter14, §235). These are the roots of the polynomial in obtained from (the §236 polynomial for odd , or the squared version for even ).
Roots of the odd- polynomial (§236)
For odd , the polynomial in can be rewritten (shifting so that , i.e. taking ) as the equation
whose roots are (source: chapter14, §236).
Euler notes that in order to express things in terms of arcs less than , one may use the identity ; this is how the examples in §237 (Example I, ; Example II, ; Example III, ) simplify the list of roots to angles between and .
Vieta’s formulas on the roots (§237)
Euler does not invoke Vieta explicitly by name, but the procedure is identical. Let the roots be . From the polynomial:
-
Sum of roots: The coefficient of (the second-highest degree) in the odd- polynomial is zero (the term is absent), so .
-
Sum of reciprocals: The ratio of the constant term to the linear coefficient gives (source: chapter14, §237). This is the partial-fraction formula for cosecant; see trig-multiple-angle-partial-fractions.
-
Product: The product of all roots equals , which after rearrangement gives the [[sine-cosine-factored-products|product formula for ]].
Roots of the cosine polynomial (§243–§244)
For the polynomial in (§243), the roots are
(source: chapter14, §243). Their sum (§244): for , the sum of all roots is zero,
For even , each positive term is paired with an equal negative term. For odd , Euler verifies case by case using (source: chapter14, §244).
Roots of the tangent equation (§249–§252)
Setting and using De Moivre:
(source: chapter14, §249). The roots of (i.e., of the numerator polynomial in when is fixed) are
(source: chapter14, §249). The sum of these roots equals (§250), and their product is determined by the constant term of the polynomial (§254).
For (odd), comparing with the equation’s highest-degree coefficient gives
Special cases (§237 Example I, II; §243–§244)
(§237):
(§237):
Why it matters
The identification of trig values at arithmetic progressions of angles as the roots of a single polynomial is the algebraic engine that generates essentially all of the identities in Chapter 14. Every partial-fraction expansion of , , etc., follows by reading one of Vieta’s formulas off the same polynomial — Euler is systematically mining a single algebraic object.