Sine and Cosine

Summary: §127 of Chapter 8. With the unit-circle convention (radius $=1$, [[pi|$\pi$]] = half the circumference), Euler names two functions of an arc $z$: $\sin z$ and $\cos z$. He fixes special values, the Pythagorean identity $(\sin z{)}^2+(\cos z{)}^2=1$, the co-function relation $\cos z=\sin (\pi /2-z)$, and the derived ratios $\tan z=\sin z/\cos z$ and $\cot z=\cos z/\sin z$. Sine and cosine are introduced as functions of arc length, not as ratios in a triangle.

Sources: chapter8 (§127)

Last updated: 2026-04-27

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Definition by arc

The radius of the circle is 1; let $z$ be an arc of this circle. Euler writes $\sin z$ for the sine of the arc $z$ and $\cos z$ for the cosine of the arc $z$ (source: chapter8, §127). On the unit circle these are exactly the perpendicular and parallel components of the radius drawn to the endpoint of the arc — the same quantities classical geometry called the half-chord and the apothem-like projection — and they coincide with the angle measure in radians, since arc length = angle on the unit circle.

This is a notational shift relative to pre-Eulerian trigonometry. Pre-Euler tables tabulated sin and cos of an angle measured in degrees, on a circle of radius typically $10^7$ for precision. Euler measures the input as an arc and uses radius 1, so $\sin z$ and $\cos z$ are pure dimensionless real numbers in $[-1,1]$ — exactly the modern conventions.

Special values

Euler tabulates:

arc $z$	$\sin z$	$\cos z$
$0$	$0$	$1$
$\pi /2$	$1$	$0$
$\pi$	$0$	$-1$
$3\pi /2$	$-1$	$0$
$2\pi$	$0$	$1$

All six values follow from the geometric interpretation on the unit circle. Periodicity with period $2\pi$ is implicit: every arc reduces modulo $2\pi$ to one in $[0,2\pi )$, and the table extends to all reals.

The Pythagorean identity

$(\sin z{)}^2+(\cos z{)}^2=1.$

Euler states this as the fundamental algebraic relation between $\sin z$ and $\cos z$ (source: chapter8, §127). It encodes the geometry: $\sin z$ and $\cos z$ are the legs of a right triangle whose hypotenuse is the radius. In Chapter 8 the identity will be repeatedly factored as

$1=(\cos z+i\sin z)(\cos z-i\sin z),$

and that complex factorization (§132) is the gateway to De Moivre’s formula and ultimately Euler’s formula.

Co-function relations

Every sine is a cosine of the complementary arc:

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

Geometrically these are reflections of the unit circle across the line $y=x$. They generalize at §128 to the full periodicity table for $\sin ((4n+k)\pi /2\pm z)$.

Sine is bounded between $-1$ and $+1$

Euler notes (§127) that “every sine and cosine lies between $+1$ and $-1$.” This is built into the unit-circle definition: the projections of a point on the unit circle onto the two axes have absolute value at most 1.

Tangent, cotangent, secant, cosecant

The four derived ratios:

$\tan z=\frac{\sin z}{\cos z},\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$

Secant and cosecant (introduced indirectly via §137) are $\sec z=1/\cos z$, $\csc z=1/\sin z$. Euler treats these as derived quantities; the primary objects are $\sin$ and $\cos$.

Why functions of arc and not angle

Two reasons surface in subsequent sections:

1. The §134 series $\sin v=v-v^3/3!+v^5/5!-\cdots$ and $\cos v=1-v^2/2!+v^4/4!-\cdots$ are clean only when the input is an arc (i.e., radians). Inputs in degrees would carry irrational conversion factors $\pi /180$ in every term.
2. The bridge to logarithms and exponentials in §138 — $e^{iv}=\cos v+i\sin v$ — likewise requires the radian convention. Other choices muddy the analysis.

The decision to measure arcs rather than angles is therefore not cosmetic. It is the decision that makes the trig series simple and Euler’s formula possible.

Related pages

• pi
• trigonometric-addition-formulas
• sine-and-cosine-series
• de-moivre-formula
• eulers-formula
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle