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Sine

Table of contents: Definitions - Illustrations - Differential equations - Specific values - Analytic properties - Symmetry and periodicity - Addition and multiplication formulas - Sums and products - Powers - Representations through other functions - Complex parts - Derivatives and integrals - Series expansions - Bounds and inequalities

Definitions

b63dce
Symbol: Sin sin(z)\sin(z) Sine

Illustrations

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Image: X-ray of sin(z)\sin(z) on z[5,5]+[5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Differential equations

21f156
sin(z)+sin(z)=0\sin''(z) + \sin(z) = 0
984d9c
y(z)+y(z)=0   where y(z)=c1sin(z)+c2cos(z)y''(z) + y(z) = 0\; \text{ where } y(z) = {c}_{1} \sin(z) + {c}_{2} \cos(z)
f1691f
y(z)+a2y(z)+b=0   where y(z)=c1sin ⁣(az)+c2cos ⁣(az)ba2y''(z) + {a}^{2} y(z) + b = 0\; \text{ where } y(z) = {c}_{1} \sin\!\left(a z\right) + {c}_{2} \cos\!\left(a z\right) - \frac{b}{{a}^{2}}

Specific values

c52772
sin(0)=0\sin(0) = 0
e2161b
sin(π)=0\sin(\pi) = 0
69c5ef
sin ⁣(π2)=1\sin\!\left(\frac{\pi}{2}\right) = 1
56667c
sin ⁣(3π2)=1\sin\!\left(\frac{3 \pi}{2}\right) = -1
3c833f
sin ⁣(π3)=32\sin\!\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
5fc688
sin ⁣(π4)=22\sin\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
ad6b74
sin ⁣(π6)=12\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}
c62afa
sin ⁣(πk)=0\sin\!\left(\pi k\right) = 0
506d0c
sin ⁣(π2+πk)=(1)k\sin\!\left(\frac{\pi}{2} + \pi k\right) = {\left(-1\right)}^{k}
09cd0b
sin(α)Q\sin(\alpha) \notin \overline{\mathbb{Q}}
713501
sin ⁣(πx)Q\sin\!\left(\pi x\right) \in \overline{\mathbb{Q}}
056c0e
(xQandsin ⁣(πx)Q)    (sin ⁣(πx){0,12,12,1,1})\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)
2f6818
zeroszC[sin(z)]={πn:nZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \left[\sin(z)\right] = \left\{ \pi n : n \in \mathbb{Z} \right\}
c5bdcc
arg maxxR[sin(x)]={π(2n+12):nZ}\mathop{\operatorname{arg\,max}}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = \left\{ \pi \left(2 n + \frac{1}{2}\right) : n \in \mathbb{Z} \right\}
ad04bd
arg minxR[sin(x)]={π(2n12):nZ}\mathop{\operatorname{arg\,min}}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = \left\{ \pi \left(2 n - \frac{1}{2}\right) : n \in \mathbb{Z} \right\}
bfe28b
maxxR[sin(x)]=1\mathop{\max}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = 1
27766c
minxR[sin(x)]=1\mathop{\min}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = -1

Analytic properties

114913
sin(z) is holomorphic on zC\sin(z) \text{ is holomorphic on } z \in \mathbb{C}
f4cc9e
poleszC{~}sin(z)={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \sin(z) = \left\{\right\}
6aa0bc
EssentialSingularities ⁣(sin(z),z,C{~})={~}\operatorname{EssentialSingularities}\!\left(\sin(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
96550d
BranchPoints ⁣(sin(z),z,C{~})={}\operatorname{BranchPoints}\!\left(\sin(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
a45c61
BranchCuts ⁣(sin(z),z,C)={}\operatorname{BranchCuts}\!\left(\sin(z), z, \mathbb{C}\right) = \left\{\right\}

Symmetry and periodicity

a2a30d
sin ⁣(z)=sin(z)\sin\!\left(-z\right) = -\sin(z)
82c83f
sin ⁣(z)=sin(z)\sin\!\left(\overline{z}\right) = \overline{\sin(z)}
6a8889
sin ⁣(z+2πk)=sin(z)\sin\!\left(z + 2 \pi k\right) = \sin(z)
393b62
sin ⁣(z+πk)=(1)ksin(z)\sin\!\left(z + \pi k\right) = {\left(-1\right)}^{k} \sin(z)
1c22f1
sin ⁣(π+z)=sin(z)\sin\!\left(\pi + z\right) = -\sin(z)
9cc0f2
sin ⁣(πz)=sin(z)\sin\!\left(\pi - z\right) = \sin(z)
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sin ⁣(π2+z)=cos(z)\sin\!\left(\frac{\pi}{2} + z\right) = \cos(z)
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sin ⁣(π2z)=cos(z)\sin\!\left(\frac{\pi}{2} - z\right) = \cos(z)

Addition and multiplication formulas

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sin ⁣(a+b)=sin(a)cos(b)+cos(a)sin(b)\sin\!\left(a + b\right) = \sin(a) \cos(b) + \cos(a) \sin(b)
508e2c
sin ⁣(ab)=sin(a)cos(b)cos(a)sin(b)\sin\!\left(a - b\right) = \sin(a) \cos(b) - \cos(a) \sin(b)
3b839c
sin ⁣(a+bi)=sin(a)cosh(b)+icos(a)sinh(b)\sin\!\left(a + b i\right) = \sin(a) \cosh(b) + i \cos(a) \sinh(b)
755655
sin ⁣(iz)=isinh(z)\sin\!\left(i z\right) = i \sinh(z)
1b11be
sin ⁣(2z)=2sin(z)cos(z)\sin\!\left(2 z\right) = 2 \sin(z) \cos(z)
729215
sin ⁣(3z)=3sin(z)4sin3 ⁣(z)\sin\!\left(3 z\right) = 3 \sin(z) - 4 \sin^{3}\!\left(z\right)
e3f8a4
sin ⁣(nz)=k=0(n1)/2(1)k(n2k+1)cosn2k1 ⁣(z)sin2k+1 ⁣(z)\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

d59bd9
sin(a)+sin(b)=2sin ⁣(a+b2)cos ⁣(ab2)\sin(a) + \sin(b) = 2 \sin\!\left(\frac{a + b}{2}\right) \cos\!\left(\frac{a - b}{2}\right)
e69cf6
sin(a)sin(b)=2cos ⁣(a+b2)sin ⁣(ab2)\sin(a) - \sin(b) = 2 \cos\!\left(\frac{a + b}{2}\right) \sin\!\left(\frac{a - b}{2}\right)
ad6c1c
sin(a)sin(b)=cos ⁣(ab)cos ⁣(a+b)2\sin(a) \sin(b) = \frac{\cos\!\left(a - b\right) - \cos\!\left(a + b\right)}{2}
012eba
sin(a)cos(b)=sin ⁣(a+b)+sin ⁣(ab)2\sin(a) \cos(b) = \frac{\sin\!\left(a + b\right) + \sin\!\left(a - b\right)}{2}
f183d0
sin(z)+cos(z)=2sin ⁣(z+π4)\sin(z) + \cos(z) = \sqrt{2} \sin\!\left(z + \frac{\pi}{4}\right)
6c3ba9
sin(z)cos(z)=2sin ⁣(zπ4)\sin(z) - \cos(z) = \sqrt{2} \sin\!\left(z - \frac{\pi}{4}\right)
adbc1a
cos(z)+isin(z)=eiz\cos(z) + i \sin(z) = {e}^{i z}
b8ab9c
k=0nsin ⁣(2ak+b)=sin ⁣(a(n+1))sin ⁣(an+b)sin(a)\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(a)}
906569
k=1n1sin ⁣(kπn)=n2n1\prod_{k=1}^{n - 1} \sin\!\left(\frac{k \pi}{n}\right) = \frac{n}{{2}^{n - 1}}

Powers

4948ea
sin2 ⁣(z)+cos2 ⁣(z)=1\sin^{2}\!\left(z\right) + \cos^{2}\!\left(z\right) = 1
954066
sin2 ⁣(z)cos2 ⁣(z)=cos ⁣(2z)\sin^{2}\!\left(z\right) - \cos^{2}\!\left(z\right) = -\cos\!\left(2 z\right)
244127
sin2 ⁣(z)=1cos2 ⁣(z)\sin^{2}\!\left(z\right) = 1 - \cos^{2}\!\left(z\right)
cf6e35
sin2 ⁣(z)=1cos ⁣(2z)2\sin^{2}\!\left(z\right) = \frac{1 - \cos\!\left(2 z\right)}{2}
acf63c
sin2 ⁣(z)=tan2 ⁣(z)1+tan2 ⁣(z)\sin^{2}\!\left(z\right) = \frac{\tan^{2}\!\left(z\right)}{1 + \tan^{2}\!\left(z\right)}
2a6702
sin3 ⁣(z)=3sin(z)sin ⁣(3z)4\sin^{3}\!\left(z\right) = \frac{3 \sin(z) - \sin\!\left(3 z\right)}{4}
54f420
sin2n ⁣(z)=14n(2nn)+24nk=0n1(1)n+k(2nk)cos ⁣(2(nk)z)\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)
71a264
sin2n+1 ⁣(z)=14nk=0n(1)n+k(2n+1k)sin ⁣((2n2k+1)z)\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)
d0505f
(cos(z)+isin(z))n=cos ⁣(nz)+isin ⁣(nz){\left(\cos(z) + i \sin(z)\right)}^{n} = \cos\!\left(n z\right) + i \sin\!\left(n z\right)
2392f5
sin2 ⁣(a)sin2 ⁣(b)=sin ⁣(a+b)sin ⁣(ab)\sin^{2}\!\left(a\right) - \sin^{2}\!\left(b\right) = \sin\!\left(a + b\right) \sin\!\left(a - b\right)
f6d0c6
sin2 ⁣(a)cos2 ⁣(b)=cos ⁣(a+b)cos ⁣(ab)\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

925e5b
sin(z)=cos ⁣(π2z)=cos ⁣(zπ2)=cos ⁣(z+π2)\sin(z) = \cos\!\left(\frac{\pi}{2} - z\right) = \cos\!\left(z - \frac{\pi}{2}\right) = -\cos\!\left(z + \frac{\pi}{2}\right)
3fb3ca
sin(z)=2tan ⁣(z2)tan2 ⁣(z2)+1\sin(z) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}
18f40c
sin(z)=eizeiz2i\sin(z) = \frac{{e}^{i z} - {e}^{-i z}}{2 i}
299209
sin(x)=Im ⁣(eix)\sin(x) = \operatorname{Im}\!\left({e}^{i x}\right)
cfc5c3
sin(z)=isinh ⁣(iz)\sin(z) = -i \sinh\!\left(i z\right)

Higher transcendental functions

54daa9
sin(z)=z0F1 ⁣(32,14z2)\sin(z) = z \,{}_0F_1\!\left(\frac{3}{2}, -\frac{1}{4} {z}^{2}\right)
0fbd15
sin(z)=πz2J1/2 ⁣(z)\sin(z) = \sqrt{\frac{\pi z}{2}} J_{1 / 2}\!\left(z\right)
d38a03
sin ⁣(πz)=πΓ(z)Γ ⁣(1z)\sin\!\left(\pi z\right) = \frac{\pi}{\Gamma(z) \Gamma\!\left(1 - z\right)}

Complex parts

729b70
Re ⁣(sin ⁣(x+iy))=sin(x)cosh(y)\operatorname{Re}\!\left(\sin\!\left(x + i y\right)\right) = \sin(x) \cosh(y)
037a6e
Im ⁣(sin ⁣(x+iy))=cos(x)sinh(y)\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos(x) \sinh(y)
abaf91
sin ⁣(x+iy)=sin2 ⁣(x)+sinh2 ⁣(y)\left|\sin\!\left(x + i y\right)\right| = \sqrt{\sin^{2}\!\left(x\right) + \sinh^{2}\!\left(y\right)}

Derivatives and integrals

f7ab32
sin(z)=cos(z)\sin'(z) = \cos(z)
297b3c
sin(z)=sin(z)\sin''(z) = -\sin(z)
612b21
sin(r)(z)=sin ⁣(z+πr2){\sin}^{(r)}(z) = \sin\!\left(z + \frac{\pi r}{2}\right)
a6667d
sin(r+2)(z)=sin(r)(z){\sin}^{(r + 2)}(z) = -{\sin}^{(r)}(z)
d81355
sin(r+4)(z)=sin(r)(z){\sin}^{(r + 4)}(z) = {\sin}^{(r)}(z)
c93b81
absin(z)dz=cos(a)cos(b)\int_{a}^{b} \sin(z) \, dz = \cos(a) - \cos(b)

Series expansions

f340cb
sin(z)=k=0(1)kz2k+1(2k+1)!\sin(z) = \sum_{k=0}^{\infty} {\left(-1\right)}^{k} \frac{{z}^{2 k + 1}}{\left(2 k + 1\right)!}
6b13be
sin ⁣(z+x)=k=0sin ⁣(z+πk2)xkk!\sin\!\left(z + x\right) = \sum_{k=0}^{\infty} \sin\!\left(z + \frac{\pi k}{2}\right) \frac{{x}^{k}}{k !}
11687b
sin(z)=zk=1(1z2π2k2)\sin(z) = z \prod_{k=1}^{\infty} \left(1 - \frac{{z}^{2}}{{\pi}^{2} {k}^{2}}\right)

Bounds and inequalities

Real arguments

4039ec
sin(x)1\left|\sin(x)\right| \le 1
c47a86
sin(x)x\left|\sin(x)\right| \le \left|x\right|
22c4f6
sin(x)4x(πx)π2\sin(x) \le \frac{4 x \left(\pi - x\right)}{{\pi}^{2}}
d38739
sin(x)x(πx)π\sin(x) \ge \frac{x \left(\pi - x\right)}{\pi}

Complex arguments

f77752
sin ⁣(x+yi)cosh(y)\left|\sin\!\left(x + y i\right)\right| \le \cosh(y)
dd5787
sin ⁣(x+yi)ey\left|\sin\!\left(x + y i\right)\right| \le {e}^{\left|y\right|}
3dd162
sin ⁣(x+yi)sinh ⁣(y)\left|\sin\!\left(x + y i\right)\right| \ge \sinh\!\left(\left|y\right|\right)
092377
sin ⁣(x+yi)y\left|\sin\!\left(x + y i\right)\right| \ge \left|y\right|
1721bf
sin(z)sinh ⁣(z)\left|\sin(z)\right| \le \sinh\!\left(\left|z\right|\right)
941a86
sin(z)<ez\left|\sin(z)\right| < {e}^{\left|z\right|}

Perturbations

f3a901
sin ⁣(x+y)sin(x)2\left|\sin\!\left(x + y\right) - \sin(x)\right| \le 2
03f713
sin ⁣(x+y)sin(x)y\left|\sin\!\left(x + y\right) - \sin(x)\right| \le \left|y\right|

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

2019-10-05 13:11:19.856591 UTC