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What is the exact value using a sum or difference formula of the expression cos 11pi over 12?
11pi/12 = pi - pi/12
cos(11pi/12) = cos(pi - pi/12)
cos(a-b) = cos(a)cos(b)+sin(a)sin(b)
cos(pi -pi/12) = cos(pi)cos(pi/12) + sin(pi)sin(pi/12)
sin(pi)=0
cos(pi)=-1
Therefore, cos(pi -pi/12) = -cos(pi/12)
pi/12=pi/3 -pi/4
cos(pi/12) = cos(pi/3 - pi/4)
= cos(pi/3)cos(pi/4)+sin(pi/3) sin(pi/4)
cos(pi/3)=1/2
sin(pi/3)=sqrt(3)/2
cos(pi/4)= sqrt(2)/2
sin(pi/4) = sqrt(2)/2
cos(pi/3)cos(pi/4)+sin(pi/3) sin(pi/4)
= (1/2)(sqrt(2)/2 ) + (sqrt(3)/2)( sqrt(2)/2)
= sqrt(2)/4 + sqrt(6) /4
= [sqrt(2)+sqrt(6)] /4
Therefore, cos(pi/12) = (sqrt(2)+sqrt(6))/4
-cos(pi/12) = -(sqrt(2)+sqrt(6))/4
cos(11pi/12) = -(sqrt(2)+sqrt(6))/4
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How do you factor this expression completely 9a4 - b2?
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