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(UP 2004,05, 08)7. If x cos α + isin α.ν cosB+isinß, thenprove that r-y-i tan-. (UP 2004. 05. 15) |
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Answer» We will use the following formulas, which can be derived from the usual addition formulas:cos x - cos y = -2 sin ((x+y)/2) sin ((x-y)/2)cos x + cos y = 2 cos ((x+y)/2) cos ((x-y)/2)sin x - sin y = 2 cos ((x+y)/2) sin ((x-y)/2)sin x + sin y = 2 sin ((x+y)/2) cos ((x-y)/2) (x - y) / (x + y)= (cos a - cos b + i sin a - i sin b) / (cos a + cos b + i sin a + i sin b)= {[-2 sin ((a+b)/2) sin ((a-b)/2)] + i [2 cos ((a+b)/2) sin ((a-b)/2)]} / {[2 cos ((a+b)/2) cos ((a-b)/2)] + i [2 sin ((a+b)/2) cos ((a-b)/2)]}Divide top and bottom by 2 cos ((a-b)/2):= {[-sin ((a+b)/2) tan ((a-b)/2)] + i [cos ((a+b)/2) tan ((a-b)/2)]} / {[cos ((a+b)/2)] + i [sin ((a+b)/2)]}= {[tan ((a-b)/2)] [-sin ((a+b)/2) + i cos ((a+b)/2)]} / [cos ((a+b)/2) + i sin ((a+b)/2)]= {[tan ((a-b)/2)] [i] [i sin ((a+b)/2) + cos ((a+b)/2)]} / [cos ((a+b)/2) + i sin ((a+b)/2)]= [tan ((a-b)/2)] [i] / 1= i tan ((a-b)/2). |
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