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Review optics: Law of refraction to Brewster's angle

abstract: \cite{2001_HRW}; see figure 34-27 on page 824
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8945218208 $\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$	no validation is available for declarations
2	0000111282: 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}.	8945218208 $\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$	9025853427 $\theta_{\rm Brewster}$	1310571337 $\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$	Not evaluated due to missing term in SymPy
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6450985774 $n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$	no validation is available for declarations
4	0000111984: 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}.	6450985774 $n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$	9029795851 $\theta_{\rm Brewster}$ 2773628333 $\theta_1$ 6353701615 $\theta_{\rm refracted}$ 7154592211 $\theta_2$	2575937347 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$	LHS diff is pdg0002941(sin(pdg0003509) - sin(pdg0004928)) RHS diff is pdg0001958(-sin(pdg0002243) + sin(pdg0007545))
5	0000111556: 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}.	2575937347 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$ 1310571337 $\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$		7696214507 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
6	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			8588429722 $\sin( 90^{\circ} - x ) = \cos( x )$	no validation is available for declarations
7	0000111886: 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}.	8588429722 $\sin( 90^{\circ} - x ) = \cos( x )$	1512581563 $x$ 7375348852 $\theta_{\rm Brewster}$	6831637424 $\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
8	0000111556: 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}.	7696214507 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$ 6831637424 $\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$		3061811650 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
9	0000111975: 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}.	3061811650 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$	7857757625 $n_1$	9756089533 $\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$	valid
10	0000111975: 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}.	9756089533 $\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$	5632428182 $\cos( \theta_{\rm Brewster} )$	2768857871 $\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$	valid
11	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			4968680693 $\tan( x ) = \frac{ \sin( x )}{\cos( x )}$	no validation is available for declarations
12	0000111886: 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}.	4968680693 $\tan( x ) = \frac{ \sin( x )}{\cos( x )}$	9906920183 $x$ 7321695558 $\theta_{\rm Brewster}$	4501377629 $\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$	valid
13	0000111556: 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}.	2768857871 $\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$ 4501377629 $\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$		3417126140 $\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$	valid
14	0000111490: 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}	3417126140 $\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$	6023986360 $x$ 5453995431 $\arctan{ x }$	8495187962 $\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$	recognized infrule but not yet supported
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8495187962 $\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$			no validation is available for declarations

Symbols used in optics: Law of refraction to Brewster's angle

0000003509: $ \theta_1 $
0000007545: $ \theta_2 $
0000004928: $ \theta_{\rm Brewster} $
0000002243: $ \theta_{\rm refracted} $
0000002941: $ n_1 $
0000001958: $ n_2 $
0000001464: $ x $

$Steps and expressions for optics: Law of refraction to Brewster's angle$

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0.006 seconds for pdg_app/to_review_derivation 4e139c9a-aaa4-44c9-a05c-70b34455f483
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0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTd2f5a5c0-1312-4d5c-8ccb-49ede8947681
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0.007 seconds for pdg_app/ 4e139c9a-aaa4-44c9-a05c-70b34455f483

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8945218208 \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)	no validation is available for declarations
2	0000111282: 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}.	8945218208 \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)	9025853427 \(\theta_{\rm Brewster}\)	1310571337 \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)	Not evaluated due to missing term in SymPy
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6450985774 \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)	no validation is available for declarations
4	0000111984: 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}.	6450985774 \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)	9029795851 \(\theta_{\rm Brewster}\) 2773628333 \(\theta_1\) 6353701615 \(\theta_{\rm refracted}\) 7154592211 \(\theta_2\)	2575937347 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\)	LHS diff is pdg0002941(sin(pdg0003509) - sin(pdg0004928)) RHS diff is pdg0001958(-sin(pdg0002243) + sin(pdg0007545))
5	0000111556: 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}.	2575937347 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\) 1310571337 \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)		7696214507 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
6	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			8588429722 \(\sin( 90^{\circ} - x ) = \cos( x )\)	no validation is available for declarations
7	0000111886: 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}.	8588429722 \(\sin( 90^{\circ} - x ) = \cos( x )\)	1512581563 \(x\) 7375348852 \(\theta_{\rm Brewster}\)	6831637424 \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
8	0000111556: 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}.	7696214507 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\) 6831637424 \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)		3061811650 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
9	0000111975: 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}.	3061811650 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)	7857757625 \(n_1\)	9756089533 \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)	valid
10	0000111975: 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}.	9756089533 \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)	5632428182 \(\cos( \theta_{\rm Brewster} )\)	2768857871 \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\)	valid
11	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			4968680693 \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)	no validation is available for declarations
12	0000111886: 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}.	4968680693 \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)	9906920183 \(x\) 7321695558 \(\theta_{\rm Brewster}\)	4501377629 \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)	valid
13	0000111556: 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}.	2768857871 \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\) 4501377629 \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)		3417126140 \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)	valid
14	0000111490: 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}	3417126140 \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)	6023986360 \(x\) 5453995431 \(\arctan{ x }\)	8495187962 \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)	recognized infrule but not yet supported
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8495187962 \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)			no validation is available for declarations