Explanation:
sin 3θ = sin(2θ + θ)
Angle sum formula:
= sin(2θ) cos θ + cos(2θ) sin θ
Double angle formulas:
= (2 sin θ cos θ) cos θ + (1 − 2 sin² θ) sin θ
Distribute:
= 2 sin θ cos² θ + sin θ − 2 sin³ θ
Pythagorean identity:
= 2 sin θ (1 − sin² θ) + sin θ − 2 sin³ θ
Distribute again:
= 2 sin θ − 2 sin³ θ + sin θ − 2 sin³ θ
Combine like terms:
= 3 sin θ − 4 sin³ θ