## review derivation: optics: Law of refraction to Brewster's angle

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Notes for this derivation:
\cite{2001_HRW}; see figure 34-27 on page 824

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
6 declare identity
1. 8588429722; locally 3940135:
$$\sin( 90^{\circ} - x ) = \cos( x )$$
$$- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 8588429722:
8588429722:
8 substitute LHS of expr 1 into expr 2
1. 6831637424; locally 7426234:
$$\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$$
$$pdg_{4928}$$
2. 7696214507; locally 4962698:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}$$
1. 3061811650; locally 9701820:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}$$
Nothing to split 6831637424:
7696214507:
3061811650:
6831637424:
7696214507:
3061811650:
15 declare final expr
1. 8495187962; locally 8186016:
$$\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$$
$$pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}$$
no validation is available for declarations 8495187962:
8495187962:
3 declare initial expr
1. 6450985774; locally 9932375:
$$n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$$
$$pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}$$
no validation is available for declarations 6450985774:
6450985774:
11 declare identity
1. 4968680693; locally 2621708:
$$\tan( x ) = \frac{ \sin( x )}{\cos( x )}$$
$$\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}$$
no validation is available for declarations 4968680693:
4968680693:
9 divide both sides by
1. 3061811650; locally 9701820:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}$$
1. 7857757625:
$$n_1$$
$$pdg_{2941}$$
1. 9756089533; locally 9314305:
$$\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$$
$$\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}$$
valid 3061811650:
9756089533:
3061811650:
9756089533:
5 substitute LHS of expr 1 into expr 2
1. 1310571337; locally 3893026:
$$\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 2575937347; locally 4176694:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}$$
1. 7696214507; locally 4962698:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}$$
Nothing to split 1310571337: no LHS/RHS split
2575937347:
7696214507:
1310571337: N/A
2575937347:
7696214507:
4 change two variables in expr
1. 6450985774; locally 9932375:
$$n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$$
$$pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}$$
1. 7154592211:
$$\theta_2$$
$$pdg_{7545}$$
2. 6353701615:
$$\theta_{\rm refracted}$$
$$pdg_{2243}$$
3. 2773628333:
$$\theta_1$$
$$pdg_{3509}$$
4. 9029795851:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
1. 2575937347; locally 4176694:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}$$
valid 6450985774:
2575937347:
6450985774:
2575937347:
12 change variable X to Y
1. 4968680693; locally 2621708:
$$\tan( x ) = \frac{ \sin( x )}{\cos( x )}$$
$$\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}$$
1. 7321695558:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 9906920183:
$$x$$
$$pdg_{1464}$$
1. 4501377629; locally 1898054:
$$\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$$
$$\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}$$
LHS diff is tan(pdg1464) - tan(pdg4928) RHS diff is tan(pdg1464) - tan(pdg4928) 4968680693:
4501377629:
4968680693:
4501377629:
13 substitute LHS of expr 1 into expr 2
1. 4501377629; locally 1898054:
$$\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$$
$$\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}$$
2. 2768857871; locally 8585856:
$$\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$$
$$\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}$$
1. 3417126140; locally 5179630:
$$\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$$
$$\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}$$
valid 4501377629:
2768857871:
3417126140:
4501377629:
2768857871:
3417126140:
1 declare initial expr
1. 8945218208; locally 5563180:
$$\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$$
$$pdg_{4928}$$
no validation is available for declarations 8945218208: no LHS/RHS split
8945218208: N/A
14 apply function to both sides of expression
1. 3417126140; locally 5179630:
$$\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$$
$$\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}$$
1. 5453995431:
$$\arctan{ x }$$
$$\operatorname{atan}{\left(pdg_{1464} \right)}$$
2. 6023986360:
$$x$$
$$pdg_{1464}$$
1. 8495187962; locally 8186016:
$$\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$$
$$pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}$$
no check performed 3417126140:
8495187962:
3417126140:
8495187962:
10 divide both sides by
1. 9756089533; locally 9314305:
$$\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$$
$$\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}$$
1. 5632428182:
$$\cos( \theta_{\rm Brewster} )$$
$$\cos{\left(pdg_{4928} \right)}$$
1. 2768857871; locally 8585856:
$$\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$$
$$\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}$$
valid 9756089533:
2768857871:
9756089533:
2768857871:
7 change variable X to Y
1. 8588429722; locally 3940135:
$$\sin( 90^{\circ} - x ) = \cos( x )$$
$$- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}$$
1. 7375348852:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 1512581563:
$$x$$
$$pdg_{1464}$$
1. 6831637424; locally 7426234:
$$\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$$
$$pdg_{4928}$$
Nothing to split 8588429722:
6831637424:
8588429722:
6831637424:
2 subtract X from both sides
1. 8945218208; locally 5563180:
$$\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$$
$$pdg_{4928}$$
1. 9025853427:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
1. 1310571337; locally 3893026:
$$\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$$
$$pdg_{4928}$$
Nothing to split 8945218208: no LHS/RHS split
1310571337: no LHS/RHS split
8945218208: N/A
1310571337: N/A
Physics Derivation Graph: Steps for optics: Law of refraction to Brewster's angle

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
$$x$$
['real'] dimensionless 140
2941 variable n_1
$$n_1$$
['real'] dimensionless index of refraction for material 1
9
4928 variable \theta_{\rm Brewster}
$$\theta_{\rm Brewster}$$
['real'] dimensionless Brewster's angle
16
7545 variable \theta_2
$$\theta_2$$
real dimensionless angle
2
1958 variable n_2
$$n_2$$
real dimensionless index of refraction for material 2
8
3509 variable \theta_1
$$\theta_1$$
real dimensionless angle
2
2243 variable \theta_{\rm refracted}
$$\theta_{\rm refracted}$$
real dimensionless refracted angle
• str_note
2
MESSAGE:
• local variable 'all_df' referenced before assignment