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How do you find the exact values of cos(13pi/24)sin(31pi/24) using the half angle formula?

Answer 1

Emery Adkins

#color(red)(cos((13π)/24)sin((13π)/24) =-1/4sqrt(2-sqrt3))#

We can start by using the double angle formula

#sin(2x) = 2sinxcosx#

#cosxsinx= 1/2sin(2x)#

∴ #cos((13π)/24)sin((13π)/24)= 1/2sin((13π)/12)#

We can use the sine half-angle identity

#sin(x/2) = ±sqrt((1 − cos x)/2)#

#sin((13π)/12) = sin(π + π/12) =-sin(π/12)#, and

#π/12 = (π/6)/2#

#π/6# is in the first quadrant, so we use the positive sign for the half-angle identity

#sin((13π)/12) = -sin((π/6)/2) = -(+sqrt((1–cos( π/6))/2))#

#sin((13π)/12)= -sqrt((1-(sqrt3)/2)/2)= -sqrt((2-sqrt3)/4)#

#sin((13π)/12) =-1/2sqrt(2-sqrt3)#

#cos((13π)/24)sin((13π)/24)= 1/2sin((13π)/12) = 1/2(-1/2sqrt(2-sqrt3))#

#cos((13π)/24)sin((13π)/24) =-1/4sqrt(2-sqrt3)#

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Answer from HIX Tutor

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