Products, quotients and powers in trigonometric form, and the exponential form

Take z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2) and multiply, remembering i² = −1: z1z2 = r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2)]. Recall the sum formulas from Secondary 2: cos(A + B) = cos A cos B − sin A sin B and sin(A + B) = sin A cos B + cos A sin B. The brackets are exactly these, so z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]: moduli multiply, amplitudes add. Multiplying z by (1/r)(cos(−θ) + i sin(−θ)) gives modulus r × (1/r) = 1 and amplitude θ − θ = 0, that is 1, so 1/z = (1/r)(cos(−θ) + i sin(−θ)), and the quotient rule follows: z1 ÷ z2 = (r1/r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)]: moduli divide, amplitudes subtract. A warning: the sum or difference of two principal amplitudes can leave ]−π, π]. Add or subtract 2π to bring the answer back. Pictures help: multiplying by z2 stretches the plane by r2 and turns it by θ2; so multiplying by i (r = 1, θ = π/2) is a quarter turn anticlockwise.