trigonometric

a) cos θ + cos 3θ + cos 5θ + cos 7θ

= (cos θ + cos 7θ) + (cos 3θ + cos 5θ)

= 2 cos 4θ cos 3θ + 2 cos 4θ cos θ

= 2 cos 4θ (cos θ + cos 3θ)

= 4 cos 4θ cos θ cos 2θ



b) cos θ + cos 3θ + cos 5θ + cos 7θ = 0

4 cos 4θ cos θ cos 2θ = 0

cos θ = 0 or cos 2θ = 0 or cos 4θ = 0

θ = π/2 or 2θ = π/2 or 2θ = 3π/2 or 4θ = π/2 or 4θ = 3π/2 or 4θ = 5π/2 or 4θ = 7π/2

θ = π/2 or π/4 or 3π/4 or π/8 or 3π/8 or 5π/8 or 7π/8

2010-10-09 21:17:05 補充：
cos θ = 0 gives θ = π/2 since 0 <= θ <= π/2

cos 2θ = 0 gives 2θ = π/2 or 3π/2 since 0 <= 2θ <= π

cos 4θ = 0 gives 4θ = π/2 or 3π/2 or 5π/2 or 9π/2 since 0 <= 4θ <= 2π