Unformatted text preview: erminal, as are Î² and Î²0 , it
follows that Î± âˆ’ Î² is coterminal with Î±0 âˆ’ Î²0 . Consider the case below where Î±0 â‰¥ Î²0 .
y y 1
P (cos(Î±0 ), sin(Î±0 ))
Î±0 âˆ’ Î²0 A(cos(Î±0 âˆ’ Î²0 ), sin(Î±0 âˆ’ Î²0 )) Q(cos(Î²0 ), sin(Î²0 )) Î±0
Î±0 âˆ’ Î² 0 Î²0
O 1 x O B (1, 0) x Since the angles P OQ and AOB are congruent, the distance between P and Q is equal to the
distance between A and B .2 The distance formula, Equation 1.1, yields
(cos(Î±0 ) âˆ’ cos(Î²0 ))2 + (sin(Î±0 ) âˆ’ sin(Î²0 ))2 = (cos(Î±0 âˆ’ Î²0 ) âˆ’ 1)2 + (sin(Î±0 âˆ’ Î²0 ) âˆ’ 0)2 Squaring both sides, we expand the left hand side of this equation as
(cos(Î±0 ) âˆ’ cos(Î²0 ))2 + (sin(Î±0 ) âˆ’ sin(Î²0 ))2 = cos2 (Î±0 ) âˆ’ 2 cos(Î±0 ) cos(Î²0 ) + cos2 (Î²0 )
+ sin2 (Î±0 ) âˆ’ 2 sin(Î±0 ) sin(Î²0 ) + sin2 (Î²0 )
= cos2 (Î±0 ) + sin2 (Î±0 ) + cos2 (Î²0 ) + sin2 (Î²0 )
âˆ’2 cos(Î±0 ) cos(Î²0 ) âˆ’ 2 sin(Î±0 ) sin(Î²0 )
From the Pythagorean Identities, cos2 (Î±0 ) + sin2 (Î±0 ) = 1 and cos2 (Î²0 ) + sin2 (Î²0 ) = 1, so
2
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, RenÃ© Descartes, Euclidean geometry

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