4.18: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/4.18.E1 4.18.E1] || [[Item:Q1669|<math>\frac{2x}{\pi} \leq \sin@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2x}{\pi} \leq \sin@@{x}</syntaxhighlight> || <math>0 \leq x, x \leq \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>(2*x)/(Pi) <= sin(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2*x,Pi] <= Sin[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
| [https://dlmf.nist.gov/4.18.E1 4.18.E1] || <math qid="Q1669">\frac{2x}{\pi} \leq \sin@@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2x}{\pi} \leq \sin@@{x}</syntaxhighlight> || <math>0 \leq x, x \leq \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>(2*x)/(Pi) <= sin(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2*x,Pi] <= Sin[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.18.E1 4.18.E1] || [[Item:Q1669|<math>\sin@@{x} \leq x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sin@@{x} \leq x</syntaxhighlight> || <math>0 \leq x, x \leq \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>sin(x) <= x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sin[x] <= x</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
| [https://dlmf.nist.gov/4.18.E1 4.18.E1] || <math qid="Q1669">\sin@@{x} \leq x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sin@@{x} \leq x</syntaxhighlight> || <math>0 \leq x, x \leq \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>sin(x) <= x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sin[x] <= x</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.18.E2 4.18.E2] || [[Item:Q1670|<math>x \leq \tan@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x \leq \tan@@{x}</syntaxhighlight> || <math>0 \leq x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>x <= tan(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x <= Tan[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
| [https://dlmf.nist.gov/4.18.E2 4.18.E2] || <math qid="Q1670">x \leq \tan@@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x \leq \tan@@{x}</syntaxhighlight> || <math>0 \leq x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>x <= tan(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x <= Tan[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.18.E3 4.18.E3] || [[Item:Q1671|<math>\cos@@{x} \leq \frac{\sin@@{x}}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cos@@{x} \leq \frac{\sin@@{x}}{x}</syntaxhighlight> || <math>0 \leq x, x \leq \pi</math> || <syntaxhighlight lang=mathematica>cos(x) <= (sin(x))/(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cos[x] <= Divide[Sin[x],x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/4.18.E3 4.18.E3] || <math qid="Q1671">\cos@@{x} \leq \frac{\sin@@{x}}{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cos@@{x} \leq \frac{\sin@@{x}}{x}</syntaxhighlight> || <math>0 \leq x, x \leq \pi</math> || <syntaxhighlight lang=mathematica>cos(x) <= (sin(x))/(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cos[x] <= Divide[Sin[x],x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.18.E3 4.18.E3] || [[Item:Q1671|<math>\frac{\sin@@{x}}{x} \leq 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@@{x}}{x} \leq 1</syntaxhighlight> || <math>0 \leq x, x \leq \pi</math> || <syntaxhighlight lang=mathematica>(sin(x))/(x) <= 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[x],x] <= 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/4.18.E3 4.18.E3] || <math qid="Q1671">\frac{\sin@@{x}}{x} \leq 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@@{x}}{x} \leq 1</syntaxhighlight> || <math>0 \leq x, x \leq \pi</math> || <syntaxhighlight lang=mathematica>(sin(x))/(x) <= 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[x],x] <= 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.18.E4 4.18.E4] || [[Item:Q1672|<math>\pi < \frac{\sin@{\pi x}}{x(1-x)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pi < \frac{\sin@{\pi x}}{x(1-x)}</syntaxhighlight> || <math>0 < x, x < 1</math> || <syntaxhighlight lang=mathematica>Pi < (sin(Pi*x))/(x*(1 - x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pi < Divide[Sin[Pi*x],x*(1 - x)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| [https://dlmf.nist.gov/4.18.E4 4.18.E4] || <math qid="Q1672">\pi < \frac{\sin@{\pi x}}{x(1-x)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pi < \frac{\sin@{\pi x}}{x(1-x)}</syntaxhighlight> || <math>0 < x, x < 1</math> || <syntaxhighlight lang=mathematica>Pi < (sin(Pi*x))/(x*(1 - x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pi < Divide[Sin[Pi*x],x*(1 - x)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.18.E4 4.18.E4] || [[Item:Q1672|<math>\frac{\sin@{\pi x}}{x(1-x)} \leq 4</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@{\pi x}}{x(1-x)} \leq 4</syntaxhighlight> || <math>0 < x, x < 1</math> || <syntaxhighlight lang=mathematica>(sin(Pi*x))/(x*(1 - x)) <= 4</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[Pi*x],x*(1 - x)] <= 4</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| [https://dlmf.nist.gov/4.18.E4 4.18.E4] || <math qid="Q1672">\frac{\sin@{\pi x}}{x(1-x)} \leq 4</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@{\pi x}}{x(1-x)} \leq 4</syntaxhighlight> || <math>0 < x, x < 1</math> || <syntaxhighlight lang=mathematica>(sin(Pi*x))/(x*(1 - x)) <= 4</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[Pi*x],x*(1 - x)] <= 4</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.18.E5 4.18.E5] || [[Item:Q1673|<math>|\sinh@@{y}| \leq |\sin@@{z}|\leq\cosh@@{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sinh@@{y}| \leq |\sin@@{z}|\leq\cosh@@{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sinh(y)) <= abs(sin(x + y*I)) <= cosh(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sinh[y]] <= Abs[Sin[x + y*I]] <= Cosh[y]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 18]
| [https://dlmf.nist.gov/4.18.E5 4.18.E5] || <math qid="Q1673">|\sinh@@{y}| \leq |\sin@@{z}|\leq\cosh@@{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sinh@@{y}| \leq |\sin@@{z}|\leq\cosh@@{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sinh(y)) <= abs(sin(x + y*I)) <= cosh(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sinh[y]] <= Abs[Sin[x + y*I]] <= Cosh[y]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 18]
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| [https://dlmf.nist.gov/4.18.E6 4.18.E6] || [[Item:Q1674|<math>|\sinh@@{y}| \leq |\cos@@{z}|\leq\cosh@@{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sinh@@{y}| \leq |\cos@@{z}|\leq\cosh@@{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sinh(y)) <= abs(cos(x + y*I)) <= cosh(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sinh[y]] <= Abs[Cos[x + y*I]] <= Cosh[y]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 18]
| [https://dlmf.nist.gov/4.18.E6 4.18.E6] || <math qid="Q1674">|\sinh@@{y}| \leq |\cos@@{z}|\leq\cosh@@{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sinh@@{y}| \leq |\cos@@{z}|\leq\cosh@@{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sinh(y)) <= abs(cos(x + y*I)) <= cosh(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sinh[y]] <= Abs[Cos[x + y*I]] <= Cosh[y]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 18]
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| [https://dlmf.nist.gov/4.18.E7 4.18.E7] || [[Item:Q1675|<math>|\csc@@{z}| \leq \csch@@{|y|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\csc@@{z}| \leq \csch@@{|y|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(csc(x + y*I)) <= csch(abs(y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Csc[x + y*I]] <= Csch[Abs[y]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
| [https://dlmf.nist.gov/4.18.E7 4.18.E7] || <math qid="Q1675">|\csc@@{z}| \leq \csch@@{|y|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\csc@@{z}| \leq \csch@@{|y|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(csc(x + y*I)) <= csch(abs(y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Csc[x + y*I]] <= Csch[Abs[y]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
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| [https://dlmf.nist.gov/4.18.E8 4.18.E8] || [[Item:Q1676|<math>|\cos@@{z}| \leq \cosh@@{|z|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\cos@@{z}| \leq \cosh@@{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(cos(z)) <= cosh(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Cos[z]] <= Cosh[Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.18.E8 4.18.E8] || <math qid="Q1676">|\cos@@{z}| \leq \cosh@@{|z|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\cos@@{z}| \leq \cosh@@{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(cos(z)) <= cosh(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Cos[z]] <= Cosh[Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.18.E9 4.18.E9] || [[Item:Q1677|<math>|\sin@@{z}| \leq \sinh@@{|z|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sin@@{z}| \leq \sinh@@{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sin(z)) <= sinh(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sin[z]] <= Sinh[Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.18.E9 4.18.E9] || <math qid="Q1677">|\sin@@{z}| \leq \sinh@@{|z|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sin@@{z}| \leq \sinh@@{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(sin(z)) <= sinh(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sin[z]] <= Sinh[Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.18#Ex1 4.18#Ex1] || [[Item:Q1678|<math>|\cos@@{z}| < 2</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\cos@@{z}| < 2</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(cos(z)) < 2</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Cos[z]] < 2</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.18#Ex1 4.18#Ex1] || <math qid="Q1678">|\cos@@{z}| < 2</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\cos@@{z}| < 2</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(cos(z)) < 2</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Cos[z]] < 2</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.18#Ex2 4.18#Ex2] || [[Item:Q1679|<math>|\sin@@{z}| \leq \tfrac{6}{5}|z|</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sin@@{z}| \leq \tfrac{6}{5}|z|</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>abs(sin(z)) <= (6)/(5)*abs(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sin[z]] <= Divide[6,5]*Abs[z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| [https://dlmf.nist.gov/4.18#Ex2 4.18#Ex2] || <math qid="Q1679">|\sin@@{z}| \leq \tfrac{6}{5}|z|</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\sin@@{z}| \leq \tfrac{6}{5}|z|</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>abs(sin(z)) <= (6)/(5)*abs(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Sin[z]] <= Divide[6,5]*Abs[z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
|}
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Latest revision as of 11:06, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.18.E1 2 ⁢ x π ≤ sin ⁡ x 2 𝑥 𝜋 𝑥 {\displaystyle{\displaystyle\frac{2x}{\pi}\leq\sin x}}
\frac{2x}{\pi} \leq \sin@@{x}		 0 ≤ x , x ≤ 1 2 ⁢ π formulae-sequence 0 𝑥 𝑥 1 2 𝜋 {\displaystyle{\displaystyle 0\leq x,x\leq\frac{1}{2}\pi}} 
(2*x)/(Pi) <= sin(x)

Divide[2*x,Pi] <= Sin[x]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
4.18.E1 sin ⁡ x ≤ x 𝑥 𝑥 {\displaystyle{\displaystyle\sin x\leq x}}
\sin@@{x} \leq x		 0 ≤ x , x ≤ 1 2 ⁢ π formulae-sequence 0 𝑥 𝑥 1 2 𝜋 {\displaystyle{\displaystyle 0\leq x,x\leq\frac{1}{2}\pi}} 
sin(x) <= x

Sin[x] <= x
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
4.18.E2 x ≤ tan ⁡ x 𝑥 𝑥 {\displaystyle{\displaystyle x\leq\tan x}}
x \leq \tan@@{x}		 0 ≤ x , x < 1 2 ⁢ π formulae-sequence 0 𝑥 𝑥 1 2 𝜋 {\displaystyle{\displaystyle 0\leq x,x<\frac{1}{2}\pi}} 
x <= tan(x)

x <= Tan[x]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
4.18.E3 cos ⁡ x ≤ sin ⁡ x x 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\cos x\leq\frac{\sin x}{x}}}
\cos@@{x} \leq \frac{\sin@@{x}}{x}		 0 ≤ x , x ≤ π formulae-sequence 0 𝑥 𝑥 𝜋 {\displaystyle{\displaystyle 0\leq x,x\leq\pi}} 
cos(x) <= (sin(x))/(x)

Cos[x] <= Divide[Sin[x],x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.18.E3 sin ⁡ x x ≤ 1 𝑥 𝑥 1 {\displaystyle{\displaystyle\frac{\sin x}{x}\leq 1}}
\frac{\sin@@{x}}{x} \leq 1		 0 ≤ x , x ≤ π formulae-sequence 0 𝑥 𝑥 𝜋 {\displaystyle{\displaystyle 0\leq x,x\leq\pi}} 
(sin(x))/(x) <= 1

Divide[Sin[x],x] <= 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.18.E4 π < sin ⁡ ( π ⁢ x ) x ⁢ ( 1 - x ) 𝜋 𝜋 𝑥 𝑥 1 𝑥 {\displaystyle{\displaystyle\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}}}
\pi < \frac{\sin@{\pi x}}{x(1-x)}		 0 < x , x < 1 formulae-sequence 0 𝑥 𝑥 1 {\displaystyle{\displaystyle 0<x,x<1}} 
Pi < (sin(Pi*x))/(x*(1 - x))

Pi < Divide[Sin[Pi*x],x*(1 - x)]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.18.E4 sin ⁡ ( π ⁢ x ) x ⁢ ( 1 - x ) ≤ 4 𝜋 𝑥 𝑥 1 𝑥 4 {\displaystyle{\displaystyle\frac{\sin\left(\pi x\right)}{x(1-x)}\leq 4}}
\frac{\sin@{\pi x}}{x(1-x)} \leq 4		 0 < x , x < 1 formulae-sequence 0 𝑥 𝑥 1 {\displaystyle{\displaystyle 0<x,x<1}} 
(sin(Pi*x))/(x*(1 - x)) <= 4

Divide[Sin[Pi*x],x*(1 - x)] <= 4
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.18.E5 | sinh ⁡ y | ≤ | sin ⁡ z | ≤ cosh ⁡ y 𝑦 𝑧 𝑦 {\displaystyle{\displaystyle|\sinh y|\leq|\sin z|\leq\cosh y}}
|\sinh@@{y}| \leq |\sin@@{z}|\leq\cosh@@{y}		 

abs(sinh(y)) <= abs(sin(x + y*I)) <= cosh(y)

Abs[Sinh[y]] <= Abs[Sin[x + y*I]] <= Cosh[y]
Failure Failure Error Successful [Tested: 18]
4.18.E6 | sinh ⁡ y | ≤ | cos ⁡ z | ≤ cosh ⁡ y 𝑦 𝑧 𝑦 {\displaystyle{\displaystyle|\sinh y|\leq|\cos z|\leq\cosh y}}
|\sinh@@{y}| \leq |\cos@@{z}|\leq\cosh@@{y}		 

abs(sinh(y)) <= abs(cos(x + y*I)) <= cosh(y)

Abs[Sinh[y]] <= Abs[Cos[x + y*I]] <= Cosh[y]
Failure Failure Error Successful [Tested: 18]
4.18.E7 | csc ⁡ z | ≤ csch ⁡ | y | 𝑧 𝑦 {\displaystyle{\displaystyle|\csc z|\leq\operatorname{csch}|y|}}
|\csc@@{z}| \leq \csch@@{|y|}		 

abs(csc(x + y*I)) <= csch(abs(y))

Abs[Csc[x + y*I]] <= Csch[Abs[y]]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
4.18.E8 | cos ⁡ z | ≤ cosh ⁡ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\cos z|\leq\cosh|z|}}
|\cos@@{z}| \leq \cosh@@{|z|}		 

abs(cos(z)) <= cosh(abs(z))

Abs[Cos[z]] <= Cosh[Abs[z]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.18.E9 | sin ⁡ z | ≤ sinh ⁡ | z | 𝑧 𝑧 {\displaystyle{\displaystyle|\sin z|\leq\sinh|z|}}
|\sin@@{z}| \leq \sinh@@{|z|}		 

abs(sin(z)) <= sinh(abs(z))

Abs[Sin[z]] <= Sinh[Abs[z]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.18#Ex1 | cos ⁡ z | < 2 𝑧 2 {\displaystyle{\displaystyle|\cos z|<2}}
|\cos@@{z}| < 2		 

abs(cos(z)) < 2

Abs[Cos[z]] < 2
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.18#Ex2 | sin ⁡ z | ≤ 6 5 ⁢ | z | 𝑧 6 5 𝑧 {\displaystyle{\displaystyle|\sin z|\leq\tfrac{6}{5}|z|}}
|\sin@@{z}| \leq \tfrac{6}{5}|z|		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
abs(sin(z)) <= (6)/(5)*abs(z)

Abs[Sin[z]] <= Divide[6,5]*Abs[z]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
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