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

Repeat the product rule on the same number: z² = r²(cos 2θ + i sin 2θ), z³ = z² · z = r³(cos 3θ + i sin 3θ), and in general zⁿ = rⁿ(cos nθ + i sin nθ) for a positive whole number n. Bring nθ back into ]−π, π] by adding or subtracting multiples of 2π if you want the principal amplitude. (Next week we give this result the name of De Moivre.) Now a new way of writing z. For every real x there are three famous sums: eˣ = 1 + x + x²/2! + x³/3! + …, cos x = 1 − x²/2! + x⁴/4! − … and sin x = x − x³/3! + x⁵/5! − … (these three are the only ones we need). Put ix in place of x in the first. Since i² = −1, i³ = −i, i⁴ = 1, i⁵ = i, the terms without i add up to 1 − x²/2! + x⁴/4! − … = cos x, and the terms with i add up to i(x − x³/3! + …) = i sin x. This is Euler’s equation: e^(ix) = cos x + i sin x, with x in radians (in Arabic صيغة أويلر, and the exponential form is الصورة الأسية). So z = r(cos θ + i sin θ) can be written z = re^(iθ). The number e^(iθ) has modulus 1, so it lies on the unit circle. Special values: e^(iπ/2) = i, e^(iπ) = −1 (so e^(iπ) + 1 = 0) and e^(2πi) = 1. Also e^(a + iθ) = eᵃ · e^(iθ) has modulus eᵃ and amplitude θ.