# Sine

## Definitions

Symbol: Sin $\sin\!\left(z\right)$ Sine

## Illustrations

Image: X-ray of $\sin\!\left(z\right)$ on $z \in \left[-5, 5\right] + \left[-5, 5\right] i$ ## Differential equations

$\sin''(z) + \sin\!\left(z\right) = 0$
$y''(z) + y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = {c}_{1} \sin\!\left(z\right) + {c}_{2} \cos\!\left(z\right)$
$y''(z) + {a}^{2} y\!\left(z\right) + b = 0\; \text{ where } y\!\left(z\right) = {c}_{1} \sin\!\left(a z\right) + {c}_{2} \cos\!\left(a z\right) - \frac{b}{{a}^{2}}$

## Specific values

$\sin\!\left(0\right) = 0$
$\sin\!\left(\pi\right) = 0$
$\sin\!\left(\frac{\pi}{2}\right) = 1$
$\sin\!\left(\frac{3 \pi}{2}\right) = -1$
$\sin\!\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
$\sin\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
$\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}$
$\sin\!\left(\pi k\right) = 0$
$\sin\!\left(\frac{\pi}{2} + \pi k\right) = {\left(-1\right)}^{k}$
$\sin\!\left(\alpha\right) \notin \overline{\mathbb{Q}}$
$\sin\!\left(\pi x\right) \in \overline{\mathbb{Q}}$
$\left(x \in \mathbb{Q} \,\mathbin{\operatorname{and}}\, \sin\!\left(\pi x\right) \in \mathbb{Q}\right) \implies \left(\sin\!\left(\pi x\right) \in \left\{0, \frac{1}{2}, -\frac{1}{2}, 1, -1\right\}\right)$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \left[\sin\!\left(z\right)\right] = \left\{ \pi n : n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{arg\,max}}\limits_{x \in \mathbb{R}} \left[\sin\!\left(x\right)\right] = \left\{ \pi \left(2 n + \frac{1}{2}\right) : n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{arg\,min}}\limits_{x \in \mathbb{R}} \left[\sin\!\left(x\right)\right] = \left\{ \pi \left(2 n - \frac{1}{2}\right) : n \in \mathbb{Z} \right\}$
$\mathop{\max}\limits_{x \in \mathbb{R}} \left[\sin\!\left(x\right)\right] = 1$
$\mathop{\min}\limits_{x \in \mathbb{R}} \left[\sin\!\left(x\right)\right] = -1$

## Analytic properties

$\operatorname{HolomorphicDomain}\!\left(\sin\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}$
$\operatorname{Poles}\!\left(\sin\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left(\sin\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(\sin\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(\sin\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}$

## Symmetry and periodicity

$\sin\!\left(-z\right) = -\sin\!\left(z\right)$
$\sin\!\left(\overline{z}\right) = \overline{\sin\!\left(z\right)}$
$\sin\!\left(z + 2 \pi k\right) = \sin\!\left(z\right)$
$\sin\!\left(z + \pi k\right) = {\left(-1\right)}^{k} \sin\!\left(z\right)$
$\sin\!\left(\pi + z\right) = -\sin\!\left(z\right)$
$\sin\!\left(\pi - z\right) = \sin\!\left(z\right)$
$\sin\!\left(\frac{\pi}{2} + z\right) = \cos\!\left(z\right)$
$\sin\!\left(\frac{\pi}{2} - z\right) = \cos\!\left(z\right)$

$\sin\!\left(a + b\right) = \sin\!\left(a\right) \cos\!\left(b\right) + \cos\!\left(a\right) \sin\!\left(b\right)$
$\sin\!\left(a - b\right) = \sin\!\left(a\right) \cos\!\left(b\right) - \cos\!\left(a\right) \sin\!\left(b\right)$
$\sin\!\left(a + b i\right) = \sin\!\left(a\right) \cosh\!\left(b\right) + i \cos\!\left(a\right) \sinh\!\left(b\right)$
$\sin\!\left(i z\right) = i \sinh\!\left(z\right)$
$\sin\!\left(2 z\right) = 2 \sin\!\left(z\right) \cos\!\left(z\right)$
$\sin\!\left(3 z\right) = 3 \sin\!\left(z\right) - 4 \sin^{3}\!\left(z\right)$
$\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)$

## Sums and products

$\sin\!\left(a\right) + \sin\!\left(b\right) = 2 \sin\!\left(\frac{a + b}{2}\right) \cos\!\left(\frac{a - b}{2}\right)$
$\sin\!\left(a\right) - \sin\!\left(b\right) = 2 \cos\!\left(\frac{a + b}{2}\right) \sin\!\left(\frac{a - b}{2}\right)$
$\sin\!\left(a\right) \sin\!\left(b\right) = \frac{\cos\!\left(a - b\right) - \cos\!\left(a + b\right)}{2}$
$\sin\!\left(a\right) \cos\!\left(b\right) = \frac{\sin\!\left(a + b\right) + \sin\!\left(a - b\right)}{2}$
$\sin\!\left(z\right) + \cos\!\left(z\right) = \sqrt{2} \sin\!\left(z + \frac{\pi}{4}\right)$
$\sin\!\left(z\right) - \cos\!\left(z\right) = \sqrt{2} \sin\!\left(z - \frac{\pi}{4}\right)$
$\cos\!\left(z\right) + i \sin\!\left(z\right) = {e}^{i z}$
$\sum_{k=0}^{n} \sin\!\left(2 a k + b\right) = \frac{\sin\!\left(a \left(n + 1\right)\right) \sin\!\left(a n + b\right)}{\sin\!\left(a\right)}$
$\prod_{k=1}^{n - 1} \sin\!\left(\frac{k \pi}{n}\right) = \frac{n}{{2}^{n - 1}}$

## Powers

$\sin^{2}\!\left(z\right) + \cos^{2}\!\left(z\right) = 1$
$\sin^{2}\!\left(z\right) - \cos^{2}\!\left(z\right) = -\cos\!\left(2 z\right)$
$\sin^{2}\!\left(z\right) = 1 - \cos^{2}\!\left(z\right)$
$\sin^{2}\!\left(z\right) = \frac{1 - \cos\!\left(2 z\right)}{2}$
$\sin^{2}\!\left(z\right) = \frac{\tan^{2}\!\left(z\right)}{1 + \tan^{2}\!\left(z\right)}$
$\sin^{3}\!\left(z\right) = \frac{3 \sin\!\left(z\right) - \sin\!\left(3 z\right)}{4}$
$\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)$
$\sin^{2 n + 1}\!\left(z\right) = \frac{1}{{4}^{n}} \sum_{k=0}^{n} {\left(-1\right)}^{n + k} {2 n + 1 \choose k} \sin\!\left(\left(2 n - 2 k + 1\right) z\right)$
${\left(\cos\!\left(z\right) + i \sin\!\left(z\right)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)$
$\sin^{2}\!\left(a\right) - \sin^{2}\!\left(b\right) = \sin\!\left(a + b\right) \sin\!\left(a - b\right)$
$\sin^{2}\!\left(a\right) - \cos^{2}\!\left(b\right) = -\cos\!\left(a + b\right) \cos\!\left(a - b\right)$

## Representations through other functions

### Elementary functions

$\sin\!\left(z\right) = \cos\!\left(\frac{\pi}{2} - z\right) = \cos\!\left(z - \frac{\pi}{2}\right) = -\cos\!\left(z + \frac{\pi}{2}\right)$
$\sin\!\left(z\right) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}$
$\sin\!\left(z\right) = \frac{{e}^{i z} - {e}^{-i z}}{2 i}$
$\sin\!\left(x\right) = \operatorname{Im}\!\left({e}^{i x}\right)$
$\sin\!\left(z\right) = -i \sinh\!\left(i z\right)$

### Higher transcendental functions

$\sin\!\left(z\right) = z \,{}_0F_1\!\left(\frac{3}{2}, -\frac{1}{4} {z}^{2}\right)$
$\sin\!\left(z\right) = \sqrt{\frac{\pi z}{2}} J_{1 / 2}\!\left(z\right)$
$\sin\!\left(\pi z\right) = \frac{\pi}{\Gamma\!\left(z\right) \Gamma\!\left(1 - z\right)}$

## Complex parts

$\operatorname{Re}\!\left(\sin\!\left(x + i y\right)\right) = \sin\!\left(x\right) \cosh\!\left(y\right)$
$\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos\!\left(x\right) \sinh\!\left(y\right)$
$\left|\sin\!\left(x + i y\right)\right| = \sqrt{\sin^{2}\!\left(x\right) + \sinh^{2}\!\left(y\right)}$

## Derivatives and integrals

$\sin'(z) = \cos\!\left(z\right)$
$\sin''(z) = -\sin\!\left(z\right)$
${\sin}^{(r)}(z) = \sin\!\left(z + \frac{\pi r}{2}\right)$
${\sin}^{(r + 2)}(z) = -{\sin}^{(r)}(z)$
${\sin}^{(r + 4)}(z) = {\sin}^{(r)}(z)$
$\int_{a}^{b} \sin\!\left(z\right) \, dz = \cos\!\left(a\right) - \cos\!\left(b\right)$

## Series expansions

$\sin\!\left(z\right) = \sum_{k=0}^{\infty} {\left(-1\right)}^{k} \frac{{z}^{2 k + 1}}{\left(2 k + 1\right)!}$
$\sin\!\left(z + x\right) = \sum_{k=0}^{\infty} \sin\!\left(z + \frac{\pi k}{2}\right) \frac{{x}^{k}}{k !}$
$\sin\!\left(z\right) = z \prod_{k=1}^{\infty} \left(1 - \frac{{z}^{2}}{{\pi}^{2} {k}^{2}}\right)$

## Bounds and inequalities

### Real arguments

$\left|\sin\!\left(x\right)\right| \le 1$
$\left|\sin\!\left(x\right)\right| \le \left|x\right|$
$\sin\!\left(x\right) \le \frac{4 x \left(\pi - x\right)}{{\pi}^{2}}$
$\sin\!\left(x\right) \ge \frac{x \left(\pi - x\right)}{\pi}$

### Complex arguments

$\left|\sin\!\left(x + y i\right)\right| \le \cosh\!\left(y\right)$
$\left|\sin\!\left(x + y i\right)\right| \le {e}^{\left|y\right|}$
$\left|\sin\!\left(x + y i\right)\right| \ge \sinh\!\left(\left|y\right|\right)$
$\left|\sin\!\left(x + y i\right)\right| \ge \left|y\right|$
$\left|\sin\!\left(z\right)\right| \le \sinh\!\left(\left|z\right|\right)$
$\left|\sin\!\left(z\right)\right| \lt {e}^{\left|z\right|}$

### Perturbations

$\left|\sin\!\left(x + y\right) - \sin\!\left(x\right)\right| \le 2$
$\left|\sin\!\left(x + y\right) - \sin\!\left(x\right)\right| \le \left|y\right|$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC