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1: 4.21 Identities

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4.21.2 $$\sin \left(u\pm v\right)=\sin u\cos v\pm \cos u\sin v,$$

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Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.16

Referenced by:

§4.21(i), (9.8.13)

Permalink:

http://dlmf.nist.gov/4.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

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4.21.17 $$2\sin u\cos v=\sin \left(u-v\right)+\sin \left(u+v\right).$$

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Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.33

Permalink:

http://dlmf.nist.gov/4.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.18 $${\sin }^{2}u-{\sin }^{2}v=\sin \left(u+v\right)\sin \left(u-v\right),$$

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Symbols:

$\sin z$: sine function

A&S Ref:

4.3.40

Permalink:

http://dlmf.nist.gov/4.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.24 $$\sin \left(-z\right)=-\sin z,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.13

Permalink:

http://dlmf.nist.gov/4.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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4.21.30 $$\sin \left(3z\right)=3\sin z-4{\sin }^{3}z,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.27

Permalink:

http://dlmf.nist.gov/4.21.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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2: 4.42 Solution of Triangles

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4.42.4 $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},$$

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Symbols:

$\sin z$: sine function, $a$: length, $b$: base, $c$: length, $A$: angle, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(ii), §4.42 and Ch.4

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4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.9 $$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$

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Symbols:

$\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.10 $$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.11 $$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$

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Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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3: 4.14 Definitions and Periodicity

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4.14.1 $$\sin z=\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}},$$

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Defines:

$\sin z$: sine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.1

Referenced by:

§4.45(ii), (9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

… ►The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. … ►

4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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5: Guide to Searching the DLMF

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Table 1: Query Examples

► ►►►►

Query	Matching records contain
…
int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
…

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… ►

Table 2: Wildcard Examples

► ►►►►►

Query	What it stands for
sin?	sin, sine, sinh, sinc, …
si$	si, sin, sine, sinh, sinc, sinInt, similar, …
…
int_$^$ sin	any definite integral of sin.

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6: 19.11 Addition Theorems

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$\sin \psi ={\displaystyle \frac{(\sin \theta \cos \varphi )\mathrm{\Delta }(\varphi )+(\sin \varphi \cos \theta )\mathrm{\Delta }(\theta )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$

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$\cos \psi ={\displaystyle \frac{\cos \theta \cos \varphi -(\sin \theta \sin \varphi )\mathrm{\Delta }(\theta )\mathrm{\Delta }(\varphi )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$

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19.11.14 $$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$

ⓘ

Defines:

$\psi$: angle (locally)

Symbols:

$\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$

Permalink:

http://dlmf.nist.gov/19.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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$$\cos \psi =(\cos \left(2\theta \right)+k^2{\sin }^4\theta )/(1-k^2{\sin }^4\theta ),$$

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$$\sin \theta =(\sin \psi )/\sqrt{(1+\cos \psi )(1+\mathrm{\Delta }(\psi ))},$$

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7: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. Schonfelder (1980) gives 40D coefficients for $\sin$, $\cos$, $\tan$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

8: 4.20 Derivatives and Differential Equations

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4.20.1 $$\frac{d}{dz}\sin z=\cos z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.105

Permalink:

http://dlmf.nist.gov/4.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.2 $$\frac{d}{dz}\cos z=-\sin z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.106

Permalink:

http://dlmf.nist.gov/4.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.7 $$\frac{d^n}{{dz}^n}\sin z=\sin \left(z+\frac{1}{2}n\pi \right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.111

Permalink:

http://dlmf.nist.gov/4.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.12 $$w=A\cos \left(az\right)+B\sin \left(az\right),$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.13 $$w=(1/a)\sin \left(az+c\right),$$

ⓘ

Symbols:

$\sin z$: sine function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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9: 10.12 Generating Function and Associated Series

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$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

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$\sin \left(z\cos \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k+1}\left(z\right)\cos \left((2k+1)\theta \right).$

… ►

$\sin z=2J_1\left(z\right)-2J_3\left(z\right)+2J_5\left(z\right)-\mathrm{\cdots },$

… ►

$\frac{1}{2}z\sin z=4J_2\left(z\right)-16J_4\left(z\right)+36J_6\left(z\right)-\mathrm{\cdots }.$

10: 4.18 Inequalities

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4.18.1 $$\frac{2x}{\pi }\le \sin x\le x,$$ $0\le x\le \frac{1}{2}\pi$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.79 4.3.80

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

… ►

4.18.3 $$\cos x\le \frac{\sin x}{x}\le 1,$$ $0\le x\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.81

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

►

4.18.4 .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.82

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.5 $$|\sinh y|\le |\sin z|\le \cosh y,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.83

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

… ►

4.18.9 $$|\sin z|\le \sinh |z|,$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.87

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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