How do you express cos(π3)sin(7π8) without using products of trigonometric functions?

1 Answer

Use Sum or Difference
cos(π3)sin(7π8)=12sin(13π24)12sin(5π24)

Explanation:

This is the derivation

sin(x+y)=sinxcosy+cosxsiny 1st equation
sin(xy)=sinxcosycosxsiny 2nd equation

subtract 2nd from the 1st

sin(x+y)sin(xy)=2cosxsiny
it follows

cosxsiny=12[sin(x+y)sin(xy)]

Now, let x=π3 and y=7π8

cos(π3)sin(7π8)=12[sin(π3+7π8)sin(π37π8)]

cos(π3)sin(7π8)=12[sin(8π+21π24)sin(8π21π24)]

cos(π3)sin(7π8)=12[sin(29π24)sin(13π24)]

cos(π3)sin(7π8)=12[sin(π+5π24)sin(13π24)]

Note: sin(13π24)=sin(13π24)

so that

cos(π3)sin(7π8)=12[sin(π+5π24)(sin(13π24))]

cos(π3)sin(7π8)=12[sin(π+5π24)+sin(13π24)]

cos(π3)sin(7π8)=
12[sinπcos(5π24)+cosπsin(5π24)+sin(13π24)]

cos(π3)sin(7π8)=12[sin(13π24)sin(5π24)]

final answer

12sin(13π24)12sin(5π24)

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