Definition:Sine
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Definition
Trigonometry
In the above right triangle, we are concerned about the angle $\theta$.
The sine of $\angle \theta$ is defined as being $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$.
Analysis
The real function $\sin: \R \to \R$ is defined as:
- $\displaystyle \sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
$\sin x$ is voiced sine (of) $x$.
Historical Note
The symbology $\sin$ was invented by William Oughtred in his 1657 work Trigonometrie.
Linguistic Note
The sine was originally written about by Āryabhaṭa, 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 referred to as jiba. Vowels in Arabic are omitted, leaving the word jb. The word jiba in Arabic is meaningless.
When Gerard of Cremona 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.
