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

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Answer 1 · 2015-08-30T17:37:03

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

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

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

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

$sin((13π)/12) =-1/2sqrt(2-sqrt3)$

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

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