Elementary Functions - 4.14 Definitions and Periodicity

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.14.E1 sin ⁑ z = e i ⁒ z - e - i ⁒ z 2 ⁒ i 𝑧 superscript 𝑒 imaginary-unit 𝑧 superscript 𝑒 imaginary-unit 𝑧 2 imaginary-unit {\displaystyle{\displaystyle\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2% \mathrm{i}}}}
\sin@@{z} = \frac{e^{\iunit z}-e^{-\iunit z}}{2\iunit}		 

sin(z) = (exp(I*z)- exp(- I*z))/(2*I)

Sin[z] == Divide[Exp[I*z]- Exp[- I*z],2*I]
Successful Successful - Successful [Tested: 7]
4.14.E2 cos ⁑ z = e i ⁒ z + e - i ⁒ z 2 𝑧 superscript 𝑒 imaginary-unit 𝑧 superscript 𝑒 imaginary-unit 𝑧 2 {\displaystyle{\displaystyle\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2}}}
\cos@@{z} = \frac{e^{\iunit z}+e^{-\iunit z}}{2}		 

cos(z) = (exp(I*z)+ exp(- I*z))/(2)

Cos[z] == Divide[Exp[I*z]+ Exp[- I*z],2]
Successful Successful - Successful [Tested: 7]
4.14.E3 cos ⁑ z + i ⁒ sin ⁑ z = e + i ⁒ z 𝑧 𝑖 𝑧 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\cos z+i\sin z=e^{+iz}}}
\cos@@{z}+ i\sin@@{z} = e^{+ iz}		 

cos(z)+ I*sin(z) = exp(+ I*z)

Cos[z]+ I*Sin[z] == Exp[+ I*z]
Successful Successful - Successful [Tested: 7]
4.14.E3 cos ⁑ z - i ⁒ sin ⁑ z = e - i ⁒ z 𝑧 𝑖 𝑧 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle\cos z-i\sin z=e^{-iz}}}
\cos@@{z}- i\sin@@{z} = e^{- iz}		 

cos(z)- I*sin(z) = exp(- I*z)

Cos[z]- I*Sin[z] == Exp[- I*z]
Successful Successful - Successful [Tested: 7]
4.14.E4 tan ⁑ z = sin ⁑ z cos ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\tan z=\frac{\sin z}{\cos z}}}
\tan@@{z} = \frac{\sin@@{z}}{\cos@@{z}}		 

tan(z) = (sin(z))/(cos(z))

Tan[z] == Divide[Sin[z],Cos[z]]
Successful Successful - Successful [Tested: 7]
4.14.E5 csc ⁑ z = 1 sin ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\csc z=\frac{1}{\sin z}}}
\csc@@{z} = \frac{1}{\sin@@{z}}		 

csc(z) = (1)/(sin(z))

Csc[z] == Divide[1,Sin[z]]
Successful Successful - Successful [Tested: 7]
4.14.E6 sec ⁑ z = 1 cos ⁑ z 𝑧 1 𝑧 {\displaystyle{\displaystyle\sec z=\frac{1}{\cos z}}}
\sec@@{z} = \frac{1}{\cos@@{z}}		 

sec(z) = (1)/(cos(z))

Sec[z] == Divide[1,Cos[z]]
Successful Successful - Successful [Tested: 7]
4.14.E7 cot ⁑ z = cos ⁑ z sin ⁑ z 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\cot z=\frac{\cos z}{\sin z}}}
\cot@@{z} = \frac{\cos@@{z}}{\sin@@{z}}		 

cot(z) = (cos(z))/(sin(z))

Cot[z] == Divide[Cos[z],Sin[z]]
Successful Successful - Successful [Tested: 7]
4.14.E7 cos ⁑ z sin ⁑ z = 1 tan ⁑ z 𝑧 𝑧 1 𝑧 {\displaystyle{\displaystyle\frac{\cos z}{\sin z}=\frac{1}{\tan z}}}
\frac{\cos@@{z}}{\sin@@{z}} = \frac{1}{\tan@@{z}}		 

(cos(z))/(sin(z)) = (1)/(tan(z))

Divide[Cos[z],Sin[z]] == Divide[1,Tan[z]]
Successful Successful - Successful [Tested: 7]
4.14.E8 sin ⁑ ( z + 2 ⁒ k ⁒ Ο€ ) = sin ⁑ z 𝑧 2 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\sin\left(z+2k\pi\right)=\sin z}}
\sin@{z+2k\pi} = \sin@@{z}		 

sin(z + 2*k*Pi) = sin(z)

Sin[z + 2*k*Pi] == Sin[z]
Successful Failure - Successful [Tested: 21]
4.14.E9 cos ⁑ ( z + 2 ⁒ k ⁒ Ο€ ) = cos ⁑ z 𝑧 2 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\cos\left(z+2k\pi\right)=\cos z}}
\cos@{z+2k\pi} = \cos@@{z}		 

cos(z + 2*k*Pi) = cos(z)

Cos[z + 2*k*Pi] == Cos[z]
Successful Failure - Successful [Tested: 21]
4.14.E10 tan ⁑ ( z + k ⁒ Ο€ ) = tan ⁑ z 𝑧 π‘˜ πœ‹ 𝑧 {\displaystyle{\displaystyle\tan\left(z+k\pi\right)=\tan z}}
\tan@{z+k\pi} = \tan@@{z}		 

tan(z + k*Pi) = tan(z)

Tan[z + k*Pi] == Tan[z]
Successful Failure - Successful [Tested: 21]
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