Question

Find the exact value using a half angle identity. Cos(-pi/8)

Answer 1

9514 1404 393

Answer:

cos(-π/8) = (√(2+√2))/2

Explanation:

The half-angle formula is ...

`\cos{(\theta)/(2)}=\pm\sqrt{(1+cos(\theta))/(2)}`

Since cosine is an even function, cos(-π/8) = cos(π/8).

For θ/2 = π/8, θ = π/4 and cos(θ) = (√2)/2. Filling in the formula, we have ...

`\cos{(-(\pi)/(8))}=\cos{(\pi)/(8)}=\sqrt{\frac{1+\cos{(\pi)/(4)}}{2}}=\sqrt{(1+(√(2))/(2))/(2)}\\\\=\sqrt{(2+√(2))/(4)}=\boxed{\frac{\sqrt{2+√(2)}}{2}}`