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review derivation: Euler equation: trigonometric relations

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
6 divide both sides by
  1. 4742644828; locally 2939484:
    \(\exp(i x)+\exp(-i x) = 2 \cos(x)\)
    \(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}\)
  1. 0004829194:
    \(2\)
    \(2\)
  1. 3829492824; locally 4383592:
    \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)
    \(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}\)
valid 4742644828:
3829492824:
4742644828:
3829492824:
8 declare final expr
  1. 4585932229; locally 4849888:
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
  2. 2103023049; locally 4849959:
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
no validation is available for declarations 4585932229:
2103023049:
4585932229:
2103023049:
11 divide both sides by
  1. 3942849294; locally 4825483:
    \(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}\)
  1. 0001921933:
    \(2 i\)
    \(2 pdg_{4621}\)
  1. 4843995999; locally 1133483:
    \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)
    \(\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}\)
valid 3942849294:
4843995999:
3942849294:
4843995999:
10 add expr 1 to expr 2
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  2. 2123139121; locally 3194924:
    \(-\exp(-i x) = -\cos(x)+i \sin(x)\)
    \(- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}\)
  1. 3942849294; locally 4825483:
    \(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}\)
valid 4938429483:
2123139121:
3942849294:
4938429483:
2123139121:
3942849294:
7 swap LHS with RHS
  1. 3829492824; locally 4383592:
    \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)
    \(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}\)
  1. 4585932229; locally 4849888:
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
valid 3829492824:
4585932229:
3829492824:
4585932229:
5 add expr 1 to expr 2
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  2. 4938429484; locally 8888883:
    \(\exp(-i x) = \cos(x)-i \sin(x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 4742644828; locally 2939484:
    \(\exp(i x)+\exp(-i x) = 2 \cos(x)\)
    \(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}\)
valid 4938429483:
4938429484:
4742644828:
4938429483:
4938429484:
4742644828:
9 multiply both sides by
  1. 4938429484; locally 8888883:
    \(\exp(-i x) = \cos(x)-i \sin(x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 0003747849:
    \(-1\)
    \(-1\)
  1. 2123139121; locally 3194924:
    \(-\exp(-i x) = -\cos(x)+i \sin(x)\)
    \(- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}\)
valid 4938429484:
2123139121:
4938429484:
2123139121:
2 change variable X to Y
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 0002393922:
    \(x\)
    \(pdg_{1464}\)
  2. 0003949052:
    \(-x\)
    \(- pdg_{1464}\)
  1. 2394853829; locally 8888881:
    \(\exp(-i x) = \cos(-x)+i \sin(-x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
valid 4938429483:
2394853829: error for dim with 2394853829
4938429483:
2394853829: N/A
12 swap LHS with RHS
  1. 4843995999; locally 1133483:
    \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)
    \(\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}\)
  1. 2103023049; locally 4849959:
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
valid 4843995999:
2103023049:
4843995999:
2103023049:
4 function is odd
  1. 4938429482; locally 8888882:
    \(\exp(-i x) = \cos(x)+i \sin(-x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 0003919391:
    \(x\)
    \(pdg_{1464}\)
  2. 0003981813:
    \(-\sin(x)\)
    \(- \sin{\left(pdg_{1464} \right)}\)
  3. 0002919191:
    \(\sin(-x)\)
    \(- \sin{\left(pdg_{1464} \right)}\)
  1. 4938429484; locally 8888883:
    \(\exp(-i x) = \cos(x)-i \sin(x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
no check performed 4938429482: error for dim with 4938429482
4938429484:
4938429482: N/A
4938429484:
1 declare initial expr
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
no validation is available for declarations 4938429483:
4938429483:
3 function is even
  1. 2394853829; locally 8888881:
    \(\exp(-i x) = \cos(-x)+i \sin(-x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 0004849392:
    \(x\)
    \(pdg_{1464}\)
  2. 0001030901:
    \(\cos(x)\)
    \(\cos{\left(pdg_{1464} \right)}\)
  3. 0003413423:
    \(\cos(-x)\)
    \(\cos{\left(pdg_{1464} \right)}\)
  1. 4938429482; locally 8888882:
    \(\exp(-i x) = \cos(x)+i \sin(-x)\)
    \(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
no check performed 2394853829: error for dim with 2394853829
4938429482: error for dim with 4938429482
2394853829: N/A
4938429482: N/A
Physics Derivation Graph: Steps for Euler equation: trigonometric relations

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)
['real']
140
4621 variable i
\(i\)
['imaginary']
imaginary unit 74
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