§4.18 Inequalities

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Keywords:: inequalities, trigonometric functions
Notes:: For (4.18.1) see Copson (1935, p. 136). (4.18.3) follows by the same method and (4.18.2) is a consequence. (4.18.5) to (4.18.9) are straightforward and (4.18.10) is obtained from the Maclaurin expansions of $\cos z$ and $\sin z$ . For the second inequality in (4.18.4), it is sufficient to show $f(x)\equiv 4x(1-x)-\sin\left(\pi x\right)\geq 0$ for $0\leq x\leq 1/2$ . The function $f(x)$ is zero at $x=0$ and $x=1/2$ and it has no zeros in $(0,1/2)$ , because $f^{\prime}(x)=4(1-2x)-\pi\cos\left(\pi x\right)$ can have only one zero in $(0,1/2)$ where $y=4(1-2x)$ intersects $y=\pi\cos\left(\pi x\right)$ . The first inequality is proved similarly.
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/4.18
See also:: Annotations for Ch.4

Jordan’s Inequality

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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: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

4.18.2		$x\leq\tan x,$
$0\leq x<\frac{1}{2}\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\tan\NVar{z}$ : tangent function and $x$ : real variable A&S Ref: 4.3.80 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E2 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

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: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

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: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

With $z=x iy$ ,

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: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

4.18.6		$\|\sinh y\|\leq\|\cos z\|\leq\cosh y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.84 Permalink: http://dlmf.nist.gov/4.18.E6 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

4.18.7		$\|\csc z\|\leq\operatorname{csch}\|y\|,$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.85 Permalink: http://dlmf.nist.gov/4.18.E7 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

4.18.8		$\|\cos z\|\leq\cosh\|z\|,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.3.86 Permalink: http://dlmf.nist.gov/4.18.E8 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

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: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

4.18.10		$\displaystyle\|\cos z\|$	$\displaystyle<2,$
		$\displaystyle\|\sin z\|$	$\displaystyle\leq\tfrac{6}{5}\|z\|$ ,
	$\|z\|<1$ .
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.88 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §4.18, §4.18 and Ch.4

For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).

4.18.10		$\displaystyle\|\cos z\|$	$\displaystyle<2,$
		$\displaystyle\|\sin z\|$	$\displaystyle\leq\tfrac{6}{5}\|z\|$ ,
	$\|z\|<1$ .
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.88 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §4.18, §4.18 and Ch.4