Pre-Calc Exam Notes 75

Pre-Calc Exam Notes 75 - Snells law gives the relation n 1...

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Sum and Difference Formulas Section 3.2 75 So since sin D = sin ( A + B + C ), we get sin A sin B + sin C sin D = sin A sin B + sin C sin ( A + B + C ) , so by Example 3.9 we get = sin A sin B + sin C (sin A cos B cos C + cos A sin B cos C + cos A cos B sin C sin A sin B sin C ) = sin A sin B + sin C sin A cos B cos C + sin C cos A sin B cos C + sin C cos A cos B sin C sin C sin A sin B sin C . It may not be immediately obvious where to go from here, but it is not completely guesswork. We need to end up with sin ( A + C ) sin ( B + C ), and we know that sin ( B + C ) = sin B cos C + cos B sin C . There are two terms involving cos B sin C , so group them together to get sin A sin B + sin C sin D = sin A sin B sin C sin A sin B sin C + sin C cos A sin B cos C + cos B sin C (sin A cos C + cos A sin C ) = sin A sin B (1 sin 2 C ) + sin C cos A sin B cos C + cos B sin C sin ( A + C ) = sin A sin B cos 2 C + sin C cos A sin B cos C + cos B sin C sin ( A + C ) . We now have two terms involving sin B cos C , which we can factor out: sin A sin B + sin C sin D = sin B cos C (sin A cos C + cos A sin C ) + cos B sin C sin ( A + C ) = sin B cos C sin ( A + C ) + cos B sin C sin ( A + C ) = sin ( A + C ) (sin B cos C + cos B sin C ) = sin ( A + C ) sin ( B + C ) Example 3.12 In the study of the propagation of electromagnetic waves,
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Unformatted text preview: Snells law gives the relation n 1 sin 1 = n 2 sin 2 , (3.20) where 1 is the angle of incidence at which a wave strikes the planar boundary between two mediums, 2 is the angle of transmission of the wave through the new medium, and n 1 and n 2 are the indexes of refraction of the two mediums. The quantity r 1 2 s = n 1 cos 1 n 2 cos 2 n 1 cos 1 + n 2 cos 2 (3.21) is called the Fresnel coefcient for normal incidence reFection of the wave for s-polarization . Show that this can be written as: r 1 2 s = sin ( 2 1 ) sin ( 2 + 1 )...
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