Almost every student learns to say “sine”, “cosine”, “leg” (or cathetus) and “hypotenuse” without knowing what these words mean. They are labels to memorise, like street names in a strange city. But each of them was once a description, and some hide unexpected stories. The sine, for example, is the result of a translation that went wrong.

Pythagoras in Latin

The most famous theorem of geometry was for a long time a sentence about real squares, drawn on the sides of a triangle. The old texts on this site wrote it in the medieval manner:

$$\textit{quadratum lateris } a \textit{ et quadratum lateris } b \textit{ aequalis quadratum lateris } c,$$

“the square of side $a$ and the square of side $b$ are equal to the square of side $c$”. With sides $3$, $4$ and $5$, the squares have areas $9$, $16$ and $25$, and indeed $9+16=25$. With $6$, $8$ and $10$: $36+64=100$. Today we write

$$a^2+b^2=c^2,$$

and the exponent $^2$ still carries in its very name the figure it came from: “squared”. The theorem is not only Greek. A Babylonian tablet from around 1800 BC, Plimpton 322, lists triples of numbers that satisfy the relation, and the Chinese book Zhoubi suanjing demonstrates the case $3$, $4$, $5$ with a figure called gougu, the Chinese names of the two shorter sides.

Cathetus and hypotenuse

The names of the sides of a right triangle come from Greek and were descriptions of position. Cathetus, the leg, comes from káthetos, “that which drops”, the same word used for a builder’s plumb line: the leg is the side that falls perpendicular to the other. Hypotenuse comes from hypoteínousa, “that which stretches underneath”: it is the side that extends beneath the right angle, from one end to the other, like the string of a bow.

The names were verbal icons: they said where each side was. Once they became labels, they lost the picture. For someone who knows Greek, “the hypotenuse is the longest side” is almost obvious; for someone who only memorised the word, it is one more rule.

The equilateral triangle and the special angles

The sine and cosine values school asks you to memorise, such as $\frac12$, $\frac{\sqrt3}{2}$ and $\frac{\sqrt2}{2}$, come from two figures. Cut an equilateral triangle of side $a$ in half along its height $h$: two right triangles appear, with angles of $30^\circ$, $60^\circ$ and $90^\circ$, hypotenuse $a$ and one leg $\frac a2$. By Pythagoras,

$$h^2+\left(\frac a2\right)^2=a^2\quad\Longrightarrow\quad h^2=\frac{3a^2}{4}\quad\Longrightarrow\quad h=\frac{a}{2}\sqrt3.$$

With the hypotenuse equal to $1$, the leg opposite $30^\circ$ measures $\frac12$ and the one opposite $60^\circ$ measures $\frac{\sqrt3}2$. Cut a square along its diagonal and the $45^\circ$ angles appear, with legs equal to $\frac{\sqrt2}2$. The table you memorise is just a reading of two figures.

The sine, a misreading

The strangest word in trigonometry has the most curious history. Around the year 500 the Indian astronomer Aryabhata used, instead of the whole chord of an arc, the half-chord, which he called ardha-jya and later simply jya: in Sanskrit, the string of a hunting bow. Arab astronomers adopted the concept and transcribed the word as jiba, which meant nothing in Arabic. Since written Arabic usually omits vowels, the word was left as just jb.

In the twelfth century, Latin translators such as Gerard of Cremona read jb as jaib, a real Arabic word meaning fold, pocket, the fold of a garment over the chest. And they translated it into Latin with the equivalent word: sinus, meaning fold, bosom, bay. Ever since, the half-chord of an arc has been called the sine, because of a vowel nobody wrote.

Degrees, minutes and seconds

The Babylonians counted in base $60$, and Greek astronomers inherited that script for angles: the full turn in $360$ degrees, each degree in $60$ parts, each part in another $60$. Ptolemy called the first ones “first small parts” and the others “second small parts”. Latin translators wrote partes minutae primae and partes minutae secundae. The first phrase shrank to minute and the second to second. When you look at a clock you are reading, without knowing it, a Babylonian notation translated from Greek into Latin.

$$1^\circ=60'\qquad 1'=60''\qquad\text{so}\qquad 1^\circ=3600''.$$

The signs $'$ and $''$ keep this count: one mark for the first division, two for the second.

Exercise: the sine of 60° in the equilateral triangle

Take an equilateral triangle of side $2$. The height splits the base into two parts of $1$, and $h^2+1^2=2^2$, so $h=\sqrt3$. In the right triangle formed, the hypotenuse is $2$, the leg opposite the $60^\circ$ angle is $\sqrt3$ and the one opposite the $30^\circ$ angle is $1$. So $\sin60^\circ=\frac{\sqrt3}2$ and $\sin30^\circ=\frac12$. Notice that $\sin^2 60^\circ+\sin^2 30^\circ=\frac34+\frac14=1$: it is Pythagoras again.

Exercise: degrees in base 60

Write $12.35^\circ$ in degrees, minutes and seconds. The whole part gives $12^\circ$. That leaves $0.35^\circ=0.35\cdot60'=21'$ exactly. So $12.35^\circ=12^\circ21'$. And $0.1^\circ$? It is $6'$, that is, $360''$. The sexagesimal script survives because $60$ has many divisors: $2$, $3$, $4$, $5$, $6$, $10$, $12$, $15$, $20$ and $30$.

Sources: Plimpton 322, MacTutor · Eli Maor, Trigonometric Delights (1998) · Florian Cajori, A History of Mathematical Notations.

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