# Definition:Sine

## Contents

## Definition

### Definition from Triangle

In the above right triangle, we are concerned about the angle $\theta$.

The **sine** of $\angle \theta$ is defined as being $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$.

### Definition from Circle

The sine of an angle in a right triangle can be extended to the full circle as follows:

Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.

Let $P = \left({x, y}\right)$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $x$-axis.

Then the **sine** of $\theta$ is defined as the length of $AP$.

### Real Numbers

The real function $\sin: \R \to \R$ is defined as:

\(\displaystyle \sin x\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots\) |

### Complex Numbers

The complex function $\sin: \C \to \C$ is defined as:

\(\displaystyle \sin z\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots\) |

## Linguistic Note

The **sine** was originally written about by Aryabhata the Elder, under the name **ardha-jyā**.

The word **jyā** is a Sanskrit word meaning **bow-string**, and in the mathematical context means the chord of a circle.

Thus the word **ardha-jyā** literally means **half-chord**. Later the first part of the word tended to be omitted, thereby leaving the word **jyā**.

When the word **jyā** was translated into Arabic, it was interpreted as **jiba**. Vowels in Arabic are omitted, leaving the word **jb**. The word **jiba** in Arabic is meaningless.

When Robert of Chester came to translate these works in the 12th century, he interpreted **jb** as the word **jaib**, meaning **pocket** or **fold** (in clothing). This he translated into Latin as **sinus** which has several meanings, of which **fold** is one, and **curve**, **winding** or **bay** are others.

Some sources credit Gerard of Cremona for this mistranslation, but it appears that he may have been following Robert, whose $1145$ translation takes precedence.

The word **sine** is pronounced the same way as the English word **sign**.

$\sin x$ is voiced **sine (of) $x$**.

## Historical Note

It was Hipparchus of Nicaea who first compiled tables relating the length of the arc of a circle and the length of the chord which gives rise to it, for various angles subtended by the arc. However, he did not actually define the concept of a sine as such.

The **sine** was first discussed by Aryabhata the Elder, under the name **ardha-jyā**, which means **half-chord**.

The the symbol $\sin$ for sine appears to have been invented by William Oughtred in his $1657$ work *Trigonometrie*, although some authors attribute it to Euler.

## Also see

- Definition:Cosine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Secant
- Definition:Cosecant

- Results about
**the sine function**can be found here.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*: Entry:**Sine** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles