How do I find the exact value of cos(-3pi/8)?

Question

How do I find the exact value of cos(-3pi/8) without using a calculator?

Answer 1

First note that cosine is a even function so cos(-3pi/8)=cos(3pi/8). In terms of angles, 3pi/8 is 67.5 degrees--which is exactly half of 135 degrees. Using the cos(x/2)= +/- sqrt((1+cos(x))/2) half angle formula, you get cos(67.5)= +/-sqrt((1-sqrt(2)/2)/2).

Answer 2

Cos 135 = - 1/sqrt(2)

cos(-3pi/8)
= Cos(3 pi/8)
= Cos[(3 pi/4) / 2]
= Cos[135 / 2]
= sqrt[(1 + Cos 135) / 2] >>Using sub-multiple angle relation and taking positive sign only as Cosine value is positive for acute angle
= sqrt[(1 - 1/sqrt{2}) /2]