Definition:Sine

Trigonometry

 * SineCosine.png

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 word sine comes from the Latin sinus which has several meanings, one of which is curve, winding or fold.

Also see

 * Cosine, tangent, cotangent, secant and cosecant.