Abstract Algebra Theory and Applications - Computer Science ...

62 CHAPTER 3 CYCLIC GROUPSanda = r cos θb = r sin θ.We sometimes abbreviate r(cos θ + i sin θ) as r cis θ. To assure that therepresentation of z is well-defined, we also require that 0 ◦ ≤ θ < 360 ◦ . Ifthe measurement is in radians, then 0 ≤ θ < 2π.Example 8. Suppose that z = 2 cis 60 ◦ . Thenanda = 2 cos 60 ◦ = 1b = 2 sin 60 ◦ = √ 3.Hence, the rectangular representation is z = 1 + √ 3 i.Conversely, if we are given a rectangular representation of a complexnumber, it is often useful to know the number’s polar representation. Ifz = 3 √ 2 − 3 √ 2 i, thenr = √ a 2 + b 2 = √ 36 = 6and( bθ = arctan = arctan(−1) = 315a)◦ ,so 3 √ 2 − 3 √ 2 i = 6 cis 315 ◦ .The polar representation of a complex number makes it easy to find productsand powers of complex numbers. The proof of the following propositionis straightforward and is left as an exercise.Proposition 3.8 Let z = r cis θ and w = s cis φ be two nonzero complexnumbers. Thenzw = rs cis(θ + φ).Example 9. If z = 3 cis(π/3) and w = 2 cis(π/6), then zw = 6 cis(π/2) =6i. Theorem 3.9 (DeMoivre) Let z = r cis θ be a nonzero complex number.Then[r cis θ] n = r n cis(nθ)for n = 1, 2, . . ..