DLMF: Untitled Document

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4.18.1

\frac{2x}{\pi}\leq\sin x\leq x,

0\leq x\leq\frac{1}{2}\pi

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : 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

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4.18.3

\cos x\leq\frac{\sin x}{x}\leq 1,

0\leq x\leq\pi

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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

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4.18.4

\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}\leq 4,

0<x<1

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : 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|\leq|\sin z|\leq\cosh y,

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : 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

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4.18.9

|\sin z|\leq\sinh|z|,

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : 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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See accompanying text — Figure 19.3.3: $F\left(\phi,k\right)$ as a function of $k^{2}$ and ${\sin}^{2}\phi$ for $-1\leq k^{2}\leq 2$ , $0\leq{\sin}^{2}\phi\leq 1$ . …If ${\sin}^{2}\phi=1/k^{2}$ ( $<1$ ), then it has the value $K\left(1/k\right)/k$ : put $c=k^{2}$ in (19.25.5) and use (19.25.1). Magnify 3D Help

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34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

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4.32.2

\sin x\cos x<\tanh x<x,

x>0

,

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Symbols:: $\cos\NVar{z}$ : cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\sin\NVar{z}$ : sine function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.32.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.32 and Ch.4

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32.11.1

w(x)=-\sqrt{\tfrac{1}{6}|x|}+d|x|^{-1/8}\sin\left(\phi(x)-\theta_{0}\right)+o% \left(|x|^{-1/8}\right),

x\to-\infty

,

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $x$ : real, $\phi(z)$ : function, $d$ : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.11(i), §32.11 and Ch.32

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32.11.6

w_{k}(x)=d|x|^{-1/4}\sin\left(\phi(x)-\theta_{0}\right)+o\left(|x|^{-1/4}% \right),

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $x$ : real, $k$ : real, $\phi(z)$ : function, $d$ : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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\Pi\left(\phi,\alpha^{2},k_{1}\right)=k\Pi\left(\beta,k^{2}\alpha^{2},k\right),

k_{1}=1/k

,

\sin\beta=k_{1}\sin\phi\leq 1

.

…

… ►provided that in the case

p=q+1

we have

|z|<1

when

|\operatorname{ph}\beta|\leq\frac{1}{2}\pi

, and

|z|<|\sin\left(\operatorname{ph}\beta\right)|

when

\frac{1}{2}\pi\leq|\operatorname{ph}\beta|\leq\pi-\delta

(

<\pi

).

… ►If

1<\alpha^{2}<\infty

, then the Cauchy principal value satisfies … ►

E\left(\phi,1\right)=\sin\phi,

… ►Let

c={\csc}^{2}\phi\neq\alpha^{2}

and

\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}

. … ►

\Pi\left(\phi,k^{2},k\right)=\frac{1}{{k^{\prime}}^{2}}\left(E\left(\phi,k% \right)-\frac{k^{2}}{\Delta}\sin\phi\cos\phi\right),

… ►

R_{C}\left(0,y\right)=0,

y<0

.

… ►

y=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin\phi)

,

0\leq\phi\leq 2\pi

.

…

… ►Throughout this subsection we assume that

0<k<1

,

0\leq\phi\leq\pi/2

, and

\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}>0

. … ►

19.9.12

\max(\sin\phi,\phi\Delta)\leq E\left(\phi,k\right)\leq\phi,

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Symbols:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$
Permalink:: http://dlmf.nist.gov/19.9.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.14

\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(% \Delta\cos\phi)^{1/3}},

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Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$
Referenced by:: §19.9(ii), §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.9.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.15

1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos% \phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin}^{2}\phi}.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$
Referenced by:: §19.9(ii), §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.9.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.16

F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{\prime}\right)\ln\left(\frac{4}{% \Delta+\cos\phi}\right)-\theta\Delta^{2},

(\sin\phi)/8<\theta<(\ln 2)/(k^{2}\sin\phi)

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\Delta$
Referenced by:: §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.9.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.9(ii), §19.9 and Ch.19

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sin ? <= ?

1—10 of 155 matching pages

1: 4.18 Inequalities

2: 19.3 Graphics

3: 34.8 Approximations for Large Parameters

4: 4.32 Inequalities

5: 32.11 Asymptotic Approximations for Real Variables

6: 19.7 Connection Formulas

7: 16.8 Differential Equations

8: 19.6 Special Cases

9: 36.4 Bifurcation Sets

10: 19.9 Inequalities