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標題:
Trigonometry
發問:
最佳解答:
sin a sin ( 2a + b) + sin b sin ( a + 2b) = cos (a + b) L.H.S. =sin a sin ( 2a+b ) + sin b sin ( a+2b ) =sin ( 2a+b ) sin a + sin ( a+2b ) sin b =(-1/2) [ cos ( 2a+b+a ) - cos ( 2a+b-a ) ] + (-1/2) [ cos ( a+2b+b ) - cos ( a+2b-b ) ] =(-1/2) [ cos ( 3a+b ) - cos ( a+b ) ] + (-1/2) [ cos ( a+3b ) - cos ( a+b ) ] =(-1/2) cos ( 3a+b ) + (1/2) cos ( a+b ) - (1/2) cos ( a+3b ) + (1/2) cos ( a+b ) =cos ( a+b ) - (1/2) cos ( 3a+b ) - (1/2) cos ( a+3b ) =cos ( a+b ) - (1/2) [ cos ( 3a+b ) + cos ( a+3b) ] =cos ( a+b ) - (1/2) { 2 cos [ ( 3a+b+a+3b )/2 ] cos [ ( 3a+b-a-3b)/2 ] } =cos ( a+b ) - cos ( 2a+2b ) cos ( a-b ) ∵a + b = 45° ∴cos ( 2a+2b ) = cos [ 2(a+b) ] = cos (2*45°) = cos 90° = 0 ∴cos ( a+b ) - cos ( 2a+2b ) cos ( a-b ) = cos ( a+b ) - 0*cos ( a-b ) = cos ( a+b ) ∴L.H.S.=R.H.S ∴sin a sin ( 2a + b) + sin b sin ( a + 2b) = cos (a + b) 2014-04-19 21:42:07 補充： Reminder： sin a cos b = 1/2 [ sin ( a+b ) + sin ( a-b ) ] sin a sin b = -1/2 [ cos ( a+b ) - cos ( a-b ) ] cos a cos b = 1/2 [ cos ( a+b ) + cos ( a-b ) ] 2014-04-19 21:43:33 補充： sin x + sin y = 2 sin [ (x+y)/2 ] cos [ (x-y)/2 ] sin x - sin y = 2 cos [ (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 sin [ (x+y)/2 ] sin [ (x-y)/2 ]
其他解答:
Trigonometry
發問:
此文章來自奇摩知識+如有不便請留言告知
If a + b = 45 degree, prove that : sin a sin ( 2a + b) + sin b sin ( a + 2b) = cos (a + b) for all real values of a and b.最佳解答:
sin a sin ( 2a + b) + sin b sin ( a + 2b) = cos (a + b) L.H.S. =sin a sin ( 2a+b ) + sin b sin ( a+2b ) =sin ( 2a+b ) sin a + sin ( a+2b ) sin b =(-1/2) [ cos ( 2a+b+a ) - cos ( 2a+b-a ) ] + (-1/2) [ cos ( a+2b+b ) - cos ( a+2b-b ) ] =(-1/2) [ cos ( 3a+b ) - cos ( a+b ) ] + (-1/2) [ cos ( a+3b ) - cos ( a+b ) ] =(-1/2) cos ( 3a+b ) + (1/2) cos ( a+b ) - (1/2) cos ( a+3b ) + (1/2) cos ( a+b ) =cos ( a+b ) - (1/2) cos ( 3a+b ) - (1/2) cos ( a+3b ) =cos ( a+b ) - (1/2) [ cos ( 3a+b ) + cos ( a+3b) ] =cos ( a+b ) - (1/2) { 2 cos [ ( 3a+b+a+3b )/2 ] cos [ ( 3a+b-a-3b)/2 ] } =cos ( a+b ) - cos ( 2a+2b ) cos ( a-b ) ∵a + b = 45° ∴cos ( 2a+2b ) = cos [ 2(a+b) ] = cos (2*45°) = cos 90° = 0 ∴cos ( a+b ) - cos ( 2a+2b ) cos ( a-b ) = cos ( a+b ) - 0*cos ( a-b ) = cos ( a+b ) ∴L.H.S.=R.H.S ∴sin a sin ( 2a + b) + sin b sin ( a + 2b) = cos (a + b) 2014-04-19 21:42:07 補充： Reminder： sin a cos b = 1/2 [ sin ( a+b ) + sin ( a-b ) ] sin a sin b = -1/2 [ cos ( a+b ) - cos ( a-b ) ] cos a cos b = 1/2 [ cos ( a+b ) + cos ( a-b ) ] 2014-04-19 21:43:33 補充： sin x + sin y = 2 sin [ (x+y)/2 ] cos [ (x-y)/2 ] sin x - sin y = 2 cos [ (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 sin [ (x+y)/2 ] sin [ (x-y)/2 ]
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