How do you express #cos(pi/ 3 ) * sin( ( 3 pi) / 8 ) # without using products of trigonometric functions?

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Answer 1 · 2016-02-14T03:17:11

#cos (pi/3)*sin((3pi)/8)=1/2*sin ((17pi)/24)+1/2*sin (pi/24)#

start with #color(red)("Sum and Difference formulas")#

#sin (x+y)=sin x cos y + cos x sin y" " " "#1st equation
#sin (x-y)=sin x cos y - cos x sin y" " " "#2nd equation

Subtract 2nd from the 1st equation

#sin (x+y)-sin (x-y)=2cos x sin y#
#2cos x sin y=sin (x+y)-sin (x-y)#

#cos x sin y =1/2 sin (x+y)-1/2 sin (x-y)#

At this point let #x=pi/3# and #y=(3pi)/8#

then use

#cos x sin y =1/2 sin (x+y)-1/2 sin (x-y)#

#cos (pi/3)*sin((3pi)/8)=1/2*sin ((17pi)/24)+1/2*sin (pi/24)#

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