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Review optics: Law of refraction to Brewster's angle

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. 8945218208
    \(\theta_{\rm Brewster} + \theta_{\rm refracted}=90^{\circ}\)
 no validation is available for declarations
2
  • 111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 8945218208
    \(\theta_{\rm Brewster} + \theta_{\rm refracted}=90^{\circ}\)
  1. 9025853427
    \(\theta_{\rm Brewster}\)
  1. 1310571337
    \(\theta_{\rm refracted}=90^{\circ} - \theta_{\rm Brewster}\)
 invalid syntax (<string>, line 0)
3
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 6450985774
    \(n_1 \sin( \theta_1 )=n_2 \sin( \theta_2 )\)
 no validation is available for declarations
4
  • 111984: change two variables in expression
  • number of inputs: 1; feeds: 4; outputs: 1
  • Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.
  1. 6450985774
    \(n_1 \sin( \theta_1 )=n_2 \sin( \theta_2 )\)
  1. 7154592211
    \(\theta_2\)
  1. 2773628333
    \(\theta_1\)
  1. 9029795851
    \(\theta_{\rm Brewster}\)
  1. 6353701615
    \(\theta_{\rm refracted}\)
  1. 2575937347
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \sin( \theta_{\rm refracted} )\)
 list index out of range
5
  • 111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 2575937347
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \sin( \theta_{\rm refracted} )\)
  1. 1310571337
    \(\theta_{\rm refracted}=90^{\circ} - \theta_{\rm Brewster}\)
  1. 7696214507
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)
 invalid syntax (<string>, line 0)
6
  • 111299: declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 8588429722
    \(\sin( 90^{\circ} - x )=\cos( x )\)
 no validation is available for declarations
7
  • 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. 8588429722
    \(\sin( 90^{\circ} - x )=\cos( x )\)
  1. 7375348852
    \(\theta_{\rm Brewster}\)
  1. 1512581563
    \(x\)
  1. 6831637424
    \(\sin( 90^{\circ} - \theta_{\rm Brewster} )=\cos( \theta_{\rm Brewster} )\)
 invalid syntax (<string>, line 0)
8
  • 111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 7696214507
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)
  1. 6831637424
    \(\sin( 90^{\circ} - \theta_{\rm Brewster} )=\cos( \theta_{\rm Brewster} )\)
  1. 3061811650
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \cos( \theta_{\rm Brewster} )\)
 invalid syntax (<string>, line 0)
9
  • 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. 3061811650
    \(n_1 \sin( \theta_{\rm Brewster} )=n_2 \cos( \theta_{\rm Brewster} )\)
  1. 7857757625
    \(n_1\)
  1. 9756089533
    \(\sin( \theta_{\rm Brewster} )=\frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)
 valid
10
  • 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. 9756089533
    \(\sin( \theta_{\rm Brewster} )=\frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)
  1. 5632428182
    \(\cos( \theta_{\rm Brewster} )\)
  1. 2768857871
    \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}=\frac{n_2}{n_1}\)
 valid
11
  • 111299: declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 4968680693
    \(\tan( x )=\frac{ \sin( x )}{\cos( x )}\)
 no validation is available for declarations
12
  • 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. 4968680693
    \(\tan( x )=\frac{ \sin( x )}{\cos( x )}\)
  1. 9906920183
    \(x\)
  1. 7321695558
    \(\theta_{\rm Brewster}\)
  1. 4501377629
    \(\tan( \theta_{\rm Brewster} )=\frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)
 valid
13
  • 111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 4501377629
    \(\tan( \theta_{\rm Brewster} )=\frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)
  1. 2768857871
    \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}=\frac{n_2}{n_1}\)
  1. 3417126140
    \(\tan( \theta_{\rm Brewster} )=\frac{ n_2 }{ n_1 }\)
 valid
14
  • 111490: apply function to both sides of expression
  • number of inputs: 1; feeds: 2; outputs: 1
  • Apply function $#1$ with argument $#2$ to Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}
  1. 3417126140
    \(\tan( \theta_{\rm Brewster} )=\frac{ n_2 }{ n_1 }\)
  1. 6023986360
    \(x\)
  1. 5453995431
    \(\arctan{ x }\)
  1. 8495187962
    \(\theta_{\rm Brewster}=\arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
 recognized infrule but not yet supported
15
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 8495187962
    \(\theta_{\rm Brewster}=\arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
 no validation is available for declarations


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