How do you find the exact values of cos(3pi/8) using the half angle formula?

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Answer 1 · 2015-08-30T18:03:49

#color(red)(cos((3π)/8) =sqrt(2–sqrt2)/2)#

The cosine half-angle formula is

#cos(x/2) = ±sqrt((1 + cos x) / 2)#

The sign is positive if #x/2# is in the first or fourth quadrant and negative if #x/2# is in the second or third quadrant.

#(3π)/8# is in the first quadrant, so the sign is positive.

#(3π)/8 = ((3π)/4)/2#

∴ #cos( (3π)/8) = cos(((3π)/4)/2) = sqrt((1+cos ((3π)/4))/2)#

#cos((3π)/8) = sqrt((1 – (sqrt2)/2)/2) = sqrt((2 – sqrt2)/4)#

#cos((3π)/8) = sqrt(2 – sqrt2)/2#