Trigonometric Addition Formulas

Summary: §128, §130, §131 of Chapter 8. Starting from the four sum/difference identities $\sin (y\pm z)=\sin y\cos z\pm \cos y\sin z$ and $\cos (y\pm z)=\cos y\cos z\mp \sin y\sin z$ — taken as known — Euler generates the entire algebraic apparatus of trigonometry: the periodicity catalog for $\sin$ and $\cos$ of $(4n+k)\pi /2\pm z$, the product-to-sum and sum-to-product theorems, the half-angle formulas, and a family of derived ratios.

Sources: chapter8 (§128, §130, §131)

Last updated: 2026-04-27

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§128 — The four base identities

Euler takes the four sum/difference formulas as part of “what is known from trigonometry”:

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

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

These are not derived in the Introductio; their geometric proof is left to standard trigonometry. Every later identity in the chapter is a consequence of these four together with the Pythagorean identity $(\sin z{)}^2+(\cos z{)}^2=1$.

§128 — The periodicity catalog

Substituting $y=\pi /2,\pi ,3\pi /2,2\pi$ produces the full quadrant-shift table:

$\sin (\pi /2+z)=+\cos z,\sin (\pi /2-z)=+\cos z,$

$\cos (\pi /2+z)=-\sin z,\cos (\pi /2-z)=+\sin z,$

$\sin (\pi +z)=-\sin z,\sin (\pi -z)=+\sin z,$

$\cos (\pi +z)=-\cos z,\cos (\pi -z)=-\cos z,$

$\sin (3\pi /2+z)=-\cos z,\sin (3\pi /2-z)=-\cos z,$

$\cos (3\pi /2+z)=+\sin z,\cos (3\pi /2-z)=-\sin z,$

$\sin (2\pi +z)=+\sin z,\sin (2\pi -z)=-\sin z,$

$\cos (2\pi +z)=+\cos z,\cos (2\pi -z)=+\cos z.$

Euler then states the general law: for any integer $n$ (positive or negative), the eight cases

$\sin (\frac{4n+k}{2}\pi \pm z),\cos (\frac{4n+k}{2}\pi \pm z),k=1,2,3,4$

reduce to $\pm \sin z$ or $\pm \cos z$ according to a $k$-mod-4 cycle (source: chapter8, §128). This is the periodicity-with-period-$2\pi$ statement, plus the half-period symmetries.

§130 — Product-to-sum

Adding and subtracting the §128 sum/difference formulas:

$\sin y\cos z=\frac{1}{2}(\sin (y+z)+\sin (y-z)),$

$\cos y\sin z=\frac{1}{2}(\sin (y+z)-\sin (y-z)),$

$\cos y\cos z=\frac{1}{2}(\cos (y-z)+\cos (y+z)),$

$\sin y\sin z=\frac{1}{2}(\cos (y-z)-\cos (y+z)).$

These convert products of trig values into sums — useful when integrating, multiplying tabulated entries, or analyzing recurrences (cf. trigonometric-recurrent-progression).

§130 — Half-angle

Setting $y=z=v/2$ in the cosine product-to-sum formulas:

$(\cos (v/2){)}^2=\frac{1+\cos v}{2},(\sin (v/2){)}^2=\frac{1-\cos v}{2},$

hence

$\cos \frac{v}{2}=\sqrt{\frac{1+\cos v}{2}},\sin \frac{v}{2}=\sqrt{\frac{1-\cos v}{2}}.$

(source: chapter8, §130). Reading the formula in reverse: knowing $\cos v$ determines $\sin (v/2)$ and $\cos (v/2)$. Iteration of this halving is the core of the §136 table-construction strategy and is the trigonometric analogue of the geometric-mean method for logarithms.

§131 — Sum-to-product

Change variables: let $a=y+z$, $b=y-z$, so $y=(a+b)/2$, $z=(a-b)/2$. Substituting in §130’s product-to-sum identities and rearranging gives the four sum-to-product theorems:

$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2},$

$\sin a-\sin b=2\cos \frac{a+b}{2}\sin \frac{a-b}{2},$

$\cos a+\cos b=2\cos \frac{a+b}{2}\cos \frac{a-b}{2},$

$\cos a-\cos b=-2\sin \frac{a+b}{2}\sin \frac{a-b}{2}.$

(source: chapter8, §131). These convert sums of sines/cosines into products — useful when factoring trigonometric polynomials and when reading off ratios.

§131 — Derived ratios

Dividing the §131 identities pairwise produces a family of ratio-and-product identities:

$\frac{\sin a+\sin b}{\sin a-\sin b}=\frac{\tan \frac{a+b}{2}}{\tan \frac{a-b}{2}},$

$\frac{\sin a+\sin b}{\cos a+\cos b}=\tan \frac{a+b}{2},\frac{\sin a+\sin b}{\cos b-\cos a}=\cot \frac{a-b}{2},$

$\frac{\sin a-\sin b}{\cos a+\cos b}=\tan \frac{a-b}{2},\frac{\sin a-\sin b}{\cos b-\cos a}=\cot \frac{a+b}{2},$

$\frac{\cos a+\cos b}{\cos b-\cos a}=\cot \frac{a+b}{2}\cot \frac{a-b}{2},$

$\frac{\sin a+\sin b}{\cos a+\cos b}=\frac{\cos b-\cos a}{\sin a-\sin b},$

$\frac{\sin a+\sin b}{\sin a-\sin b}\cdot \frac{\cos a+\cos b}{\cos b-\cos a}={\cot }^2\frac{a-b}{2},$

$\frac{\sin a+\sin b}{\sin a-\sin b}\cdot \frac{\cos b-\cos a}{\cos a+\cos b}={\tan }^2\frac{a+b}{2}.$

Each is a one-line consequence of §131. Euler’s purpose in writing them all out is mostly catalogue: every standard identity of high-school trigonometry is on the table for use in later chapters.

How these formulas are used in Chapter 8

• §129 uses the §128 sum/difference formulas to derive the recurrent-progression structure of $\sin (ky+z)$ and $\cos (ky+z)$.
• §136 uses the §131 sum-to-product formulas (with $y=\pi /6$) to extend a table of sines and cosines from arcs $\le 30°$ to arcs in $[30°,60°]$, and hence to all arcs by periodicity.
• §137 uses $\tan (2a)=2\tan a/(1-{\tan }^{2}a)$ (a one-line consequence of §128 applied to $\tan =\sin /\cos$) for the same extension on tangents and cotangents.

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