This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.
| Index |
Inference Rule |
Input latex |
Feeds latex |
Output latex |
step validity |
dimension check |
unit check |
notes |
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6
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divide both sides by |
-
4742644828; locally 2939484:
\(\exp(i x)+\exp(-i x) = 2 \cos(x)\)
\(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}\)
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-
0004829194:
\(2\)
\(2\)
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-
3829492824; locally 4383592:
\(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)
\(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}\)
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valid |
4742644828:
3829492824:
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4742644828:
3829492824:
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8
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declare final expr |
-
4585932229; locally 4849888:
\(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
\(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
-
2103023049; locally 4849959:
\(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
\(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
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no validation is available for declarations |
4585932229:
2103023049:
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4585932229:
2103023049:
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11
|
divide both sides by |
-
3942849294; locally 4825483:
\(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}\)
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-
0001921933:
\(2 i\)
\(2 pdg_{4621}\)
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-
4843995999; locally 1133483:
\(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)
\(\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}\)
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valid |
3942849294:
4843995999:
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3942849294:
4843995999:
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10
|
add expr 1 to expr 2 |
-
4938429483; locally 8888888:
\(\exp(i x) = \cos(x)+i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
-
2123139121; locally 3194924:
\(-\exp(-i x) = -\cos(x)+i \sin(x)\)
\(- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}\)
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|
-
3942849294; locally 4825483:
\(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}\)
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valid |
4938429483:
2123139121:
3942849294:
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4938429483:
2123139121:
3942849294:
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7
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swap LHS with RHS |
-
3829492824; locally 4383592:
\(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)
\(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}\)
|
|
-
4585932229; locally 4849888:
\(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
\(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
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valid |
3829492824:
4585932229:
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3829492824:
4585932229:
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5
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add expr 1 to expr 2 |
-
4938429483; locally 8888888:
\(\exp(i x) = \cos(x)+i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
-
4938429484; locally 8888883:
\(\exp(-i x) = \cos(x)-i \sin(x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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|
-
4742644828; locally 2939484:
\(\exp(i x)+\exp(-i x) = 2 \cos(x)\)
\(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}\)
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valid |
4938429483:
4938429484:
4742644828:
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4938429483:
4938429484:
4742644828:
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9
|
multiply both sides by |
-
4938429484; locally 8888883:
\(\exp(-i x) = \cos(x)-i \sin(x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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-
0003747849:
\(-1\)
\(-1\)
|
-
2123139121; locally 3194924:
\(-\exp(-i x) = -\cos(x)+i \sin(x)\)
\(- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}\)
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valid |
4938429484:
2123139121:
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4938429484:
2123139121:
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2
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change variable X to Y |
-
4938429483; locally 8888888:
\(\exp(i x) = \cos(x)+i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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-
0002393922:
\(x\)
\(pdg_{1464}\)
-
0003949052:
\(-x\)
\(- pdg_{1464}\)
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-
2394853829; locally 8888881:
\(\exp(-i x) = \cos(-x)+i \sin(-x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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valid |
4938429483:
2394853829:
error for dim with 2394853829
|
4938429483:
2394853829:
N/A
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12
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swap LHS with RHS |
-
4843995999; locally 1133483:
\(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)
\(\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}\)
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-
2103023049; locally 4849959:
\(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
\(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
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valid |
4843995999:
2103023049:
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4843995999:
2103023049:
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4
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function is odd |
-
4938429482; locally 8888882:
\(\exp(-i x) = \cos(x)+i \sin(-x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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-
0003919391:
\(x\)
\(pdg_{1464}\)
-
0003981813:
\(-\sin(x)\)
\(- \sin{\left(pdg_{1464} \right)}\)
-
0002919191:
\(\sin(-x)\)
\(- \sin{\left(pdg_{1464} \right)}\)
|
-
4938429484; locally 8888883:
\(\exp(-i x) = \cos(x)-i \sin(x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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no check performed |
4938429482:
error for dim with 4938429482
4938429484:
|
4938429482:
N/A
4938429484:
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1
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declare initial expr |
|
|
-
4938429483; locally 8888888:
\(\exp(i x) = \cos(x)+i \sin(x)\)
\(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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no validation is available for declarations |
4938429483:
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4938429483:
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3
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function is even |
-
2394853829; locally 8888881:
\(\exp(-i x) = \cos(-x)+i \sin(-x)\)
\(e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
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-
0004849392:
\(x\)
\(pdg_{1464}\)
-
0001030901:
\(\cos(x)\)
\(\cos{\left(pdg_{1464} \right)}\)
-
0003413423:
\(\cos(-x)\)
\(\cos{\left(pdg_{1464} \right)}\)
|
-
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
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Physics Derivation Graph: Steps for Euler equation: trigonometric relations
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