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

Aug 30, 2015

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

Explanation:

The cosine half-angle formula is

cos(x/2) = ±sqrt((1 + cos x) / 2)

The sign is positive if $\frac{x}{2}$ is in the first or fourth quadrant and negative if $\frac{x}{2}$ 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