Definition:Sine

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


 * $$\sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$$

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

Geometry


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

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

Historical Note
The symbology $$\sin$$ was invented by William Oughtred in his 1657 work Trigonometrie.

Also see

 * Cosine, tangent, cotangent, secant and cosecant.