Posted by tony on Sunday, June 16, 2013 at 11:20pm.
cos A = 1 / 3
sin A = + OR - sqrt ( 1 - cos A ^ 2 )
sin A = + OR - sqrt ( 1 - ( 1 / 3 ) ^ 2 )
sin A = + OR - sqrt ( 1 - 1 / 9 )
sin A = + OR - sqrt ( 9 / 9 - 1 / 9 )
sin A = + OR - sqrt ( 8 / 9 )
sin A = + OR - sqrt ( 4 * 2 / 9 )
sin A = + OR - sqrt ( 4 ) * sqrt ( 2 ) / sqrt ( 9 )
sin A = + OR - 2 * sqrt ( 2 ) / 3
sin A = + OR - ( 2 / 3 ) * sqrt ( 2 )
In quadrant I sine are positive so :
sin A = ( 2 / 3 ) * sqrt ( 2 )
sin B = - 1 / 2
sin B = + OR - sqrt ( 1- cos B ^ 2 )
cos B = + OR - sqrt ( 1 - ( - 1 / 2 ) ^ 2 )
cos B = + OR - sqrt ( 1 - 1 / 4 )
cos B = + OR - sqrt ( 4 / 4 - 1 / 4 )
cos B = + OR - sqrt ( 3 / 4 )
cos B = + OR - sqrt ( 3 ) / 2
In quadrant IV cosine are positive so :
cos B = sqrt ( 3 ) / 2
cos ( A + B ) = cos A * cos B - sin A * sin B
cos ( A + B ) = ( 1 / 3 ) * sqrt ( 3 ) / 2 - ( 2 / 3 ) * sqrt ( 2 ) * ( - 1 / 2 )
cos ( A + B ) = ( 1 / 6 )sqrt ( 3 ) + ( 2 / 6 ) * sqrt ( 2 )
cos ( A + B ) = ( 1 / 6 ) [ sqrt ( 3 ) + 2 sqrt ( 2 ) ]
cos ( A + B ) = [ sqrt ( 3 ) + 2 sqrt ( 2 ) ] / 6
what's all this work?
A is in QI, so sinA = √8/3
B is in QIV so cosB = √3/2
and then as done in the final paragraph