anonymous 5 years ago cos(3π/8) Without a calculator. has to be exact…

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 $$\cos(3\pi/8)=\cos((\pi/8)+(\pi/4))=\cos(\pi/8)\cos(\pi/4)-\sin(\pi/8)\sin(\pi/4)$$.