COSMOS Chapter 3 Solution 15 - Edubirdie

University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2024
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477

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1
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diamundgalx45p

COSMOS: Complete Online Solutions Manual Organization System Chapter 3, Solution 15. Note: B = B ( cos β i + sin β j) B′ = B ( cos β i − sin β j) C = C ( cos α i + sin α j) By definition: B × C = BC sin (α − β ) (1) B′ × C = BC sin (α + β ) (2) Now ... B × C = B ( cos β i + sin β j) × C ( cos α i + sin α j) = BC ( cos β sin α − sin β cos α ) k and (3) B′ × C = B ( cos β i − sin β j) × C ( cos α i + sin α j) = BC ( cos β sin α + sin β cos α ) k (4) Equating the magnitudes of B × C from equations (1) and (3) yields: BC sin (α − β ) = BC ( cos β sin α − sin β cos α ) (5) Similarly, equating the magnitudes of B′ × C from equations (2) and (4) yields: BC sin (α + β ) = BC ( cos β sin α + sin β cos α ) (6) Adding equations (5) and (6) gives: sin (α − β ) + sin (α + β ) = 2cos β sin α or . sin α cos β = 1 1 sin (α + β ) + sin (α − β ) 2 2

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