How do you express #sin(pi/ 4 ) * cos( ( 3 pi) / 4 ) # without using products of trigonometric functions?

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Answer 1 · 2016-02-14T03:29:23

#sin (pi/4) cos ((3pi)/4) =1/2 sin (pi)+1/2 sin ((-pi)/2))=-1/2#

start with #color(blue)("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

Add 1st and 2nd equations

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

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

At this point let #x=pi/4# and #y=(3pi)/4#

then use

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

#sin (pi/4) cos ((3pi)/4) =1/2 sin (pi/4+(3pi)/4)+1/2 sin (pi/4-(3pi)/4)#

#sin (pi/4) cos ((3pi)/4) =1/2 sin (pi)+1/2 sin ((-pi)/2))#

#sin (pi/4) cos ((3pi)/4) =1/2*( 0)+1/2 *(-1)#

#sin (pi/4) cos ((3pi)/4) =-1/2#

have a nice day from the Philippines ..