# Angle Sine Function

A Angle Sine Function is a trigonometric function and a periodic function with period $2\pi$

• Context:
• It can be defined as the ratio between lengths of the opposite side ($a$) to the acute angle $\theta$ and the hypotenuse ($h$) in a right triangle, $\sin(\theta)= \frac{a}{h}$.
• It can be defined as the imaginary part of the complex exponential function $\sin (\theta) = Im\left[e^{i\theta}\right]$
• It can also be defined as by the following power series, for any real number ($x$), $\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$.
• It can also be represented as a generalized continued fraction $\sin (x) = \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 + \cfrac{2\cdot3 x^2}{4\cdot5-x^2 + \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.$.
• It must satisfy the following properties, where $cos(x)$ is the cosine function, $\binom nk$ is the binomial coefficient, $\Gamma(x)$ is the gamma function, C is a constant, x and y are real numbers:
1. $\sin^2 (x) + \cos^2 (x) = 1 \quad$ Pythagorean identity
2. $\sin(\theta)=\sin(\theta+ 2\pi k)\quad$ Periodic Function
3. $\sin\left(x+y\right)=\sin x \cos y + \cos x \sin y\quad$ Sum
4. $\sin\left(x-y\right)=\sin x \cos y - \cos x \sin \quad$ Difference
5. $\sin\left(2x\right)= 2 \sin x \cos x\quad$ Double-angle formula
6. $\sin\left(nx\right)= \sum_{k=0}^n\binom nk \cos^k x\;\sin^{n-k} x\;\sin(\frac{1}{2}(n-k)\pi) \quad$ multiple-angle formula
7. $\frac{d}{dx}\sin(x) = \cos(x)\quad$ Derivative
8. $\int\sin(x)\;\mathrm{d}x = -\cos(x)+C\quad$ Indefinitive integral
9. $\int_0^\infty \sin(x^n)\;\mathrm{d}x = \Gamma(1+\frac{1}{n})sin\left(\frac{\pi}{2n}\right)\quad$ Definitive integral
• Example(s):
• $\sin(\theta) = \cos\left(\pi/2 - \theta \right)$ , where $cos(x)$ is the cosine function
• $\sin(\theta) = \pm\sqrt{1 - \cos^2(\theta)}$
• $\sin(\theta) = 1 / \csc(\theta)$, where $\csc(x)$ is the cosecant function
• $\sin(\theta) = (e^{i\theta}-e^{-i\theta})/2i=\sinh(i\theta)/i$, where $i$ is the imaginary number and $sinh(x)$ is the hyperbolic sine function
• $\sin(\theta)= \pm\frac{1}{\sqrt{1 + \cot^2(\theta)}}$, where $\cot(x)$ is the cotangent function
• $\sin(\theta) = \pm\frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}}$ , where $tan(x)$ is the tangent function
• $\sin(\theta)= \pm\frac{\sqrt{\sec^2(\theta) - 1}}{\sec(\theta)}$ , where $sec(x)$ is the secant function
• Counter-Example(s):
• See: Cosine Function, Cosecant Function, Cotangent Function, Secant Function, Tangent Function, Complex Exponential Function, Hyperbolic Sine Function, Pythagorean Theorem, Pythagorean identity

## References

### 1999

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/Sine.html
• QUOTE: The sine function $sin\;x$ is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let $\theta$ be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then $sin\;\theta$ is the vertical coordinate of the arc endpoint, as illustrated in the left figure above [1].

The common schoolbook definition of the sine of an angle theta in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e., :$sin\;\theta=opposite/hypotenuse$