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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

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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

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sin⁡′′(z)+sin⁡(z)=0\sin''(z) + \sin(z) = 0
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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)
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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

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sin⁡(0)=0\sin(0) = 0
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sin⁡(π)=0\sin(\pi) = 0
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sin⁡ ⁣(π2)=1\sin\!\left(\frac{\pi}{2}\right) = 1
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sin⁡ ⁣(3π2)=−1\sin\!\left(\frac{3 \pi}{2}\right) = -1
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sin⁡ ⁣(π3)=32\sin\!\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
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sin⁡ ⁣(π4)=22\sin\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
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sin⁡ ⁣(π6)=12\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}
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sin⁡ ⁣(πk)=0\sin\!\left(\pi k\right) = 0
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sin⁡ ⁣(π2+πk)=(−1)k\sin\!\left(\frac{\pi}{2} + \pi k\right) = {\left(-1\right)}^{k}
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sin⁡(α)∉Q‾\sin(\alpha) \notin \overline{\mathbb{Q}}
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sin⁡ ⁣(πx)∈Q‾\sin\!\left(\pi x\right) \in \overline{\mathbb{Q}}
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(x∈Q  and⁡  sin⁡ ⁣(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\; \sin\!\left(\pi x\right) \in \left\{0, \frac{1}{2}, -\frac{1}{2}, 1, -1\right\}
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zeros⁡ z∈C[sin⁡(z)]={πn:n∈Z}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \left[\sin(z)\right] = \left\{ \pi n : n \in \mathbb{Z} \right\}
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arg max⁡x∈R[sin⁡(x)]={π(2n+12):n∈Z}\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\}
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arg min⁡x∈R[sin⁡(x)]={π(2n−12):n∈Z}\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\}
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max⁡x∈R[sin⁡(x)]=1\mathop{\max}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = 1
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min⁡x∈R[sin⁡(x)]=−1\mathop{\min}\limits_{x \in \mathbb{R}} \left[\sin(x)\right] = -1

Analytic properties

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sin⁡(z) is holomorphic on z∈C\sin(z) \text{ is holomorphic on } z \in \mathbb{C}
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poles⁡ z∈C∪{∞~}sin⁡(z)={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \sin(z) = \left\{\right\}
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EssentialSingularities⁡ ⁣(sin⁡(z),z,C∪{∞~})={∞~}\operatorname{EssentialSingularities}\!\left(\sin(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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BranchPoints⁡ ⁣(sin⁡(z),z,C∪{∞~})={}\operatorname{BranchPoints}\!\left(\sin(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts⁡ ⁣(sin⁡(z),z,C)={}\operatorname{BranchCuts}\!\left(\sin(z), z, \mathbb{C}\right) = \left\{\right\}

Symmetry and periodicity

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sin⁡ ⁣(−z)=−sin⁡(z)\sin\!\left(-z\right) = -\sin(z)
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sin⁡ ⁣(z‾)=sin⁡(z)‾\sin\!\left(\overline{z}\right) = \overline{\sin(z)}
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sin⁡ ⁣(z+2πk)=sin⁡(z)\sin\!\left(z + 2 \pi k\right) = \sin(z)
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sin⁡ ⁣(z+πk)=(−1)ksin⁡(z)\sin\!\left(z + \pi k\right) = {\left(-1\right)}^{k} \sin(z)
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sin⁡ ⁣(π+z)=−sin⁡(z)\sin\!\left(\pi + z\right) = -\sin(z)
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sin⁡ ⁣(π−z)=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⁡ ⁣(π2−z)=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)
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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)
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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)
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sin⁡ ⁣(iz)=isinh⁡(z)\sin\!\left(i z\right) = i \sinh(z)
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sin⁡ ⁣(2z)=2sin⁡(z)cos⁡(z)\sin\!\left(2 z\right) = 2 \sin(z) \cos(z)
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sin⁡ ⁣(3z)=3sin⁡(z)−4sin⁡3 ⁣(z)\sin\!\left(3 z\right) = 3 \sin(z) - 4 \sin^{3}\!\left(z\right)
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sin⁡ ⁣(nz)=∑k=0⌊(n−1)/2⌋(−1)k(n2k+1)cos⁡n−2k−1 ⁣(z)sin⁡2k+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

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sin⁡(a)+sin⁡(b)=2sin⁡ ⁣(a+b2)cos⁡ ⁣(a−b2)\sin(a) + \sin(b) = 2 \sin\!\left(\frac{a + b}{2}\right) \cos\!\left(\frac{a - b}{2}\right)
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sin⁡(a)−sin⁡(b)=2cos⁡ ⁣(a+b2)sin⁡ ⁣(a−b2)\sin(a) - \sin(b) = 2 \cos\!\left(\frac{a + b}{2}\right) \sin\!\left(\frac{a - b}{2}\right)
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sin⁡(a)sin⁡(b)=cos⁡ ⁣(a−b)−cos⁡ ⁣(a+b)2\sin(a) \sin(b) = \frac{\cos\!\left(a - b\right) - \cos\!\left(a + b\right)}{2}
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sin⁡(a)cos⁡(b)=sin⁡ ⁣(a+b)+sin⁡ ⁣(a−b)2\sin(a) \cos(b) = \frac{\sin\!\left(a + b\right) + \sin\!\left(a - b\right)}{2}
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sin⁡(z)+cos⁡(z)=2sin⁡ ⁣(z+π4)\sin(z) + \cos(z) = \sqrt{2} \sin\!\left(z + \frac{\pi}{4}\right)
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sin⁡(z)−cos⁡(z)=2sin⁡ ⁣(z−π4)\sin(z) - \cos(z) = \sqrt{2} \sin\!\left(z - \frac{\pi}{4}\right)
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cos⁡(z)+isin⁡(z)=eiz\cos(z) + i \sin(z) = {e}^{i z}
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∑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)}
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∏k=1n−1sin⁡ ⁣(kπn)=n2n−1\prod_{k=1}^{n - 1} \sin\!\left(\frac{k \pi}{n}\right) = \frac{n}{{2}^{n - 1}}

Powers

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sin⁡2 ⁣(z)+cos⁡2 ⁣(z)=1\sin^{2}\!\left(z\right) + \cos^{2}\!\left(z\right) = 1
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sin⁡2 ⁣(z)−cos⁡2 ⁣(z)=−cos⁡ ⁣(2z)\sin^{2}\!\left(z\right) - \cos^{2}\!\left(z\right) = -\cos\!\left(2 z\right)
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sin⁡2 ⁣(z)=1−cos⁡2 ⁣(z)\sin^{2}\!\left(z\right) = 1 - \cos^{2}\!\left(z\right)
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sin⁡2 ⁣(z)=1−cos⁡ ⁣(2z)2\sin^{2}\!\left(z\right) = \frac{1 - \cos\!\left(2 z\right)}{2}
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sin⁡2 ⁣(z)=tan⁡2 ⁣(z)1+tan⁡2 ⁣(z)\sin^{2}\!\left(z\right) = \frac{\tan^{2}\!\left(z\right)}{1 + \tan^{2}\!\left(z\right)}
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sin⁡3 ⁣(z)=3sin⁡(z)−sin⁡ ⁣(3z)4\sin^{3}\!\left(z\right) = \frac{3 \sin(z) - \sin\!\left(3 z\right)}{4}
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sin⁡2n ⁣(z)=14n(2nn)+24n∑k=0n−1(−1)n+k(2nk)cos⁡ ⁣(2(n−k)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)
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sin⁡2n+1 ⁣(z)=14n∑k=0n(−1)n+k(2n+1k)sin⁡ ⁣((2n−2k+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)
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(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)
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sin⁡2 ⁣(a)−sin⁡2 ⁣(b)=sin⁡ ⁣(a+b)sin⁡ ⁣(a−b)\sin^{2}\!\left(a\right) - \sin^{2}\!\left(b\right) = \sin\!\left(a + b\right) \sin\!\left(a - b\right)
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sin⁡2 ⁣(a)−cos⁡2 ⁣(b)=−cos⁡ ⁣(a+b)cos⁡ ⁣(a−b)\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

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sin⁡(z)=cos⁡ ⁣(π2−z)=cos⁡ ⁣(z−π2)=−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)
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sin⁡(z)=2tan⁡ ⁣(z2)tan⁡2 ⁣(z2)+1\sin(z) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}
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sin⁡(z)=eiz−e−iz2i\sin(z) = \frac{{e}^{i z} - {e}^{-i z}}{2 i}
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sin⁡(x)=Im⁡ ⁣(eix)\sin(x) = \operatorname{Im}\!\left({e}^{i x}\right)
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sin⁡(z)=−isinh⁡ ⁣(iz)\sin(z) = -i \sinh\!\left(i z\right)

Higher transcendental functions

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sin⁡(z)=z 0F1 ⁣(32,−14z2)\sin(z) = z \,{}_0F_1\!\left(\frac{3}{2}, -\frac{1}{4} {z}^{2}\right)
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sin⁡(z)=πz2J1/2 ⁣(z)\sin(z) = \sqrt{\frac{\pi z}{2}} J_{1 / 2}\!\left(z\right)
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sin⁡ ⁣(πz)=πΓ(z)Γ ⁣(1−z)\sin\!\left(\pi z\right) = \frac{\pi}{\Gamma(z) \Gamma\!\left(1 - z\right)}

Complex parts

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

Derivatives and integrals

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sin⁡′(z)=cos⁡(z)\sin'(z) = \cos(z)
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sin⁡′′(z)=−sin⁡(z)\sin''(z) = -\sin(z)
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sin⁡(r)(z)=sin⁡ ⁣(z+πr2){\sin}^{(r)}(z) = \sin\!\left(z + \frac{\pi r}{2}\right)
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sin⁡(r+2)(z)=−sin⁡(r)(z){\sin}^{(r + 2)}(z) = -{\sin}^{(r)}(z)
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sin⁡(r+4)(z)=sin⁡(r)(z){\sin}^{(r + 4)}(z) = {\sin}^{(r)}(z)
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∫absin⁡(z) dz=cos⁡(a)−cos⁡(b)\int_{a}^{b} \sin(z) \, dz = \cos(a) - \cos(b)

Series expansions

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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)!}
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sin⁡ ⁣(z+x)=∑k=0∞sin⁡ ⁣(z+πk2)xkk!\sin\!\left(z + x\right) = \sum_{k=0}^{\infty} \sin\!\left(z + \frac{\pi k}{2}\right) \frac{{x}^{k}}{k !}
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sin⁡(z)=z∏k=1∞(1−z2π2k2)\sin(z) = z \prod_{k=1}^{\infty} \left(1 - \frac{{z}^{2}}{{\pi}^{2} {k}^{2}}\right)

Bounds and inequalities

Real arguments

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∣sin⁡(x)∣≤1\left|\sin(x)\right| \le 1
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∣sin⁡(x)∣≤∣x∣\left|\sin(x)\right| \le \left|x\right|
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sin⁡(x)≤4x(π−x)π2\sin(x) \le \frac{4 x \left(\pi - x\right)}{{\pi}^{2}}
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sin⁡(x)≥x(π−x)π\sin(x) \ge \frac{x \left(\pi - x\right)}{\pi}

Complex arguments

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∣sin⁡ ⁣(x+yi)∣≤cosh⁡(y)\left|\sin\!\left(x + y i\right)\right| \le \cosh(y)
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∣sin⁡ ⁣(x+yi)∣≤e∣y∣\left|\sin\!\left(x + y i\right)\right| \le {e}^{\left|y\right|}
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∣sin⁡ ⁣(x+yi)∣≥sinh⁡ ⁣(∣y∣)\left|\sin\!\left(x + y i\right)\right| \ge \sinh\!\left(\left|y\right|\right)
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∣sin⁡ ⁣(x+yi)∣≥∣y∣\left|\sin\!\left(x + y i\right)\right| \ge \left|y\right|
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∣sin⁡(z)∣≤sinh⁡ ⁣(∣z∣)\left|\sin(z)\right| \le \sinh\!\left(\left|z\right|\right)
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∣sin⁡(z)∣<e∣z∣\left|\sin(z)\right| < {e}^{\left|z\right|}

Perturbations

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∣sin⁡ ⁣(x+y)−sin⁡(x)∣≤2\left|\sin\!\left(x + y\right) - \sin(x)\right| \le 2
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∣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.

2021-03-15 19:12:00.328586 UTC