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Review Euler equation: trigonometric relations

step inference rule input feed output step validity (as per SymPy)
1
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 4938429483
    \(\exp(i x)=\cos(x)+i \sin(x)\)
 no validation is available for declarations
2
  • 111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.
  1. 4938429483
    \(\exp(i x)=\cos(x)+i \sin(x)\)
  1. 0003949052
    \(-x\)
  1. 0002393922
    \(x\)
  1. 2394853829
    \(\exp(-i x)=\cos(-x)+i \sin(-x)\)
 LHS diff is 2*sinh(pdg0001464*pdg0004621) RHS diff is 2*pdg0004621*sin(pdg0001464)
3
  • 111329: function is even
  • number of inputs: 1; feeds: 3; outputs: 1
  • $#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}.
  1. 2394853829
    \(\exp(-i x)=\cos(-x)+i \sin(-x)\)
  1. 0003413423
    \(\cos(-x)\)
  1. 0001030901
    \(\cos(x)\)
  1. 0004849392
    \(x\)
  1. 4938429482
    \(\exp(-i x)=\cos(x)+i \sin(-x)\)
 recognized infrule but not yet supported
4
  • 111522: function is odd
  • number of inputs: 1; feeds: 3; outputs: 1
  • $#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}.
  1. 4938429482
    \(\exp(-i x)=\cos(x)+i \sin(-x)\)
  1. 0003919391
    \(x\)
  1. 0002919191
    \(\sin(-x)\)
  1. 0003981813
    \(-\sin(x)\)
  1. 4938429484
    \(\exp(-i x)=\cos(x)-i \sin(x)\)
 recognized infrule but not yet supported
5
  • 111980: add expr 1 to expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Add Eq.~\\ref{eq:#1} to Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#2}.
  1. 4938429483
    \(\exp(i x)=\cos(x)+i \sin(x)\)
  1. 4938429484
    \(\exp(-i x)=\cos(x)-i \sin(x)\)
  1. 4742644828
    \(\exp(i x)+\exp(-i x)=2 \cos(x)\)
 valid
6
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 4742644828
    \(\exp(i x)+\exp(-i x)=2 \cos(x)\)
  1. 0004829194
    \(2\)
  1. 3829492824
    \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)=\cos(x)\)
 valid
7
  • 111268: swap LHS with RHS
  • number of inputs: 1; feeds: 0; outputs: 1
  • Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.
  1. 3829492824
    \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)=\cos(x)\)
  1. 4585932229
    \(\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
 valid
8
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 2103023049
    \(\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 4585932229
    \(\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
 no validation is available for declarations
9
  • 111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 4938429484
    \(\exp(-i x)=\cos(x)-i \sin(x)\)
  1. 0003747849
    \(-1\)
  1. 2123139121
    \(-\exp(-i x)=-\cos(x)+i \sin(x)\)
 valid
10
  • 111980: add expr 1 to expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Add Eq.~\\ref{eq:#1} to Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#2}.
  1. 2123139121
    \(-\exp(-i x)=-\cos(x)+i \sin(x)\)
  1. 4938429483
    \(\exp(i x)=\cos(x)+i \sin(x)\)
  1. 3942849294
    \(\exp(i x)-\exp(-i x)=2 i \sin(x)\)
 valid
11
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 3942849294
    \(\exp(i x)-\exp(-i x)=2 i \sin(x)\)
  1. 0001921933
    \(2 i\)
  1. 4843995999
    \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right)=\sin(x)\)
 valid
12
  • 111268: swap LHS with RHS
  • number of inputs: 1; feeds: 0; outputs: 1
  • Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.
  1. 4843995999
    \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right)=\sin(x)\)
  1. 2103023049
    \(\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
 valid


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