Question

Use an appropriate half-angle formula to find the exact value of the expression. cos 3π 8

Answer 1

The half-angle formula for cosine is
$\cos{\left(\dfrac{\theta}{2}\right)}=\sqrt{\dfrac{1+\cos{\theta}}{2}}$

To find cos(3π/8), we can use this formula with θ=3π/4. Cos(3π/4) = -(√2)/2, so we have

$\cos{\left(\dfrac{3\pi}{8}\right)=\sqrt{\dfrac{1-\frac{\sqrt{2}}{2}}{2}}\\\\=\sqrt{\dfrac{2-\sqrt{2}}{4}}=\frac{1}{2}\sqrt{2-\sqrt{2}}$