Question

cos(3π/8) Without a calculator. has to be exact…

Answer 1

First we need to know \(\cos(\pi/8)\). Let \(x=\pi/8\).
\(\cos(2x)=2\cos^2(x)-1\)
On the other hand
\(\cos(2x)=\cos(\pi/4)=\frac{1}{2}\sqrt{2}\)
It follows that
\(2\cos^2(x)-1=\frac{1}{2}\sqrt{2}\)
\(\cos x=\frac{1}{2}\sqrt{1+\frac{1}{2}\sqrt{2}}\).
Then compute \(cos(3\pi/8)\) by using
\(\cos(3\pi/8)=\cos((\pi/8)+(\pi/4))=\cos(\pi/8)\cos(\pi/4)-\sin(\pi/8)\sin(\pi/4)\).