Question #48b47

1 Answer
Shiva Prakash M V
Feb 13, 2018

Thus, the sum
#cos750#degrees#+sin(-17pi/16)=sqrt3/2+sqrt(2sqrt2-sqrt(1+2sqrt2))/(4sqrt2)#

Explanation:

#cos750degrees=cos(720+30)=cos(2*360+30)=cos30deg=(sqrt3)/2#
#cos750# degrees #=sqrt3/2#
#pi radians = 180 degrees#
#17pi/16=(16+1)/16pi=pi+pi/16#
#sin((-17pi)/16)=-sin(17pi/16)#
#sin(17pi/16)=sin(pi+pi/16)=-sin(pi/16)#
Hence,
#sin(-17pi/16)=sin(pi/16)#
#sin(pi/16)=sin(1/2(pi/8))#
#sinA=sqrt((1-cos2A)/2)#
#A=pi/16, 2a=pi/8#
Thus
#sin(pi/16)=sqrt((1-cos(pi/8))/2)#
#cosA=sqrt((1+cos2A)/2)#

#For A=pi/8, 2A=pi/4#
#cos(pi/8)=sqrt((1+cos(pi/4))/2)#
#cos(pi/4)=1/sqrt2#
#cos(pi/8)=sqrt((1+1/sqrt2)/2)#
#cos(pi/8)=sqrt(1+2sqrt2)/(2sqrt2)#
#sin(pi/16)=sqrt((1-sqrt(1+2sqrt2)/(2sqrt2))/2)#
#sin(pi/16)=sqrt(2sqrt2-sqrt(1+2sqrt2))/(4sqrt2)#
Now,

#cos750#degrees#=sqrt3/2#
and
#sin(-17pi/16)=sqrt(2sqrt2-sqrt(1+2sqrt2))/(4sqrt2)#

Thus, the sum
#cos750#degrees#+sin(-17pi/16)=sqrt3/2+sqrt(2sqrt2-sqrt(1+2sqrt2))/(4sqrt2)#

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