Polar form and De Moivre's theorem

Polar form, arguments, multiplication and division in polar form and De Moivre's theorem. 7 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Polar form

Write in polar form \(r\operatorname{cis}\theta\) (principal argument).

a) \(z = \sqrt3 + i\) b) \(w = -1 + i\) c) \(u = -4\sqrt3 - 4i\) d) \(v = 2 - 2\sqrt3 i\) e) \(p = -6i\)

Show solution

a) \(2\operatorname{cis}\frac{\pi}{6}\)
b) \(\sqrt2\operatorname{cis}\frac{3\pi}{4}\)
c) \(8\operatorname{cis}\frac{7\pi}{6}\)
d) \(4\operatorname{cis}\frac{5\pi}{3}\)
e) \(6\operatorname{cis}\frac{3\pi}{2}\)

Problem 2 Standard form

Write in the form \(a + bi\).

a) \(4\operatorname{cis}\frac{\pi}{3}\) b) \(\operatorname{cis}\frac{5\pi}{4}\) c) \(3(\cos 180^\circ + i\sin 180^\circ)\) d) \(2\sqrt2\operatorname{cis}225^\circ\) e) \(\frac12\operatorname{cis}\frac{11\pi}{6}\)

Show solution

a) \(2 + 2\sqrt3 i\)
b) \(-\frac{\sqrt2}{2} - \frac{\sqrt2}{2}i\)
c) \(-3\)
d) \(-2 - 2i\)
e) \(\frac{\sqrt3}{4} - \frac14 i\)

Problem 3 Arguments

a) Find \(\operatorname{Arg}(z)\) for \(z = \sin 50^\circ - i\cos 50^\circ\).

b) Find \(\operatorname{Arg}(z)\) for \(z = -\sin 20^\circ + i\cos 20^\circ\).

c) Find \(\operatorname{Arg}(z)\) for \(z = 1 + \cos\theta + i\sin\theta\), \(0 < \theta < \pi\).

d) \(z \ne 0\) and \(z^2 + |z|^2 = 0\). Find \(\operatorname{Arg}(z)\).

Show solution

a) \(\sin 50^\circ = \cos(-40^\circ)\), \(-\cos 50^\circ = \sin(-40^\circ)\): \(\operatorname{Arg} = 320^\circ\).

b) \(-\sin 20^\circ = \cos 110^\circ\) and \(\cos 20^\circ = \sin 110^\circ\): \(\operatorname{Arg} = 110^\circ\).

c) \(z = 2\cos^2\frac\theta2 + 2i\sin\frac\theta2\cos\frac\theta2 = 2\cos\frac\theta2\operatorname{cis}\frac\theta2\): \(\operatorname{Arg} = \frac\theta2\).

d) \(z = a + bi\): \(a^2 - b^2 + 2abi + a^2 + b^2 = 2a^2 + 2abi = 0\), so \(a = 0\): \(z\) is pure imaginary, \(\operatorname{Arg} = \frac{\pi}{2}\) or \(\frac{3\pi}{2}\).

Problem 4 Multiplication and division

a) \(z_1 = 6\operatorname{cis}100^\circ\), \(z_2 = 3\operatorname{cis}40^\circ\). Write \(z_1z_2\) and \(\frac{z_1}{z_2}\) as \(a + bi\).

b) Calculate \([5(\cos 20^\circ + i\sin 20^\circ)]\cdot[2(\cos 70^\circ + i\sin 70^\circ)]\).

c) Calculate \(\dfrac{8(\cos 25^\circ + i\sin 25^\circ)}{2(\cos 115^\circ + i\sin 115^\circ)}\).

d) \(z_1 = \operatorname{cis}80^\circ\), \(z_2 = \operatorname{cis}20^\circ\). Write \(z_1 + z_2\) and \(z_1 - z_2\) as \(a + bi\).

Show solution

a) \(18\operatorname{cis}140^\circ \approx -13.79 + 11.57i\); \(2\operatorname{cis}60^\circ = 1 + \sqrt3 i\).

b) \(10\operatorname{cis}90^\circ = 10i\).

c) \(4\operatorname{cis}(-90^\circ) = -4i\).

d) \(\cos 80^\circ + \cos 20^\circ = 2\cos 50^\circ\cos 30^\circ\), \(\sin 80^\circ + \sin 20^\circ = 2\sin 50^\circ\cos 30^\circ\): \(z_1 + z_2 = \sqrt3\operatorname{cis}50^\circ \approx 1.113 + 1.327i\). \(z_1 - z_2 = \operatorname{cis}140^\circ \approx -0.766 + 0.643i\).

Problem 5 Using arguments

a) \(|z_1| = |z_2| = 3\), \(\operatorname{Arg}(z_1z_2) = 100^\circ\) and \(\operatorname{Arg}\left(\frac{z_1}{z_2}\right) = 40^\circ\). Find \(z_1\) and \(z_2\).

b) \(z_1 = \cos 50^\circ - i\sin 50^\circ\), \(z_2 = \sin 70^\circ + i\cos 70^\circ\), \(z_3 = \cos 10^\circ + i\sin 10^\circ\). Calculate \(\frac{z_1z_2}{z_3}\).

Show solution

a) \(\theta_1 + \theta_2 = 100^\circ\), \(\theta_1 - \theta_2 = 40^\circ\): \(\theta_1 = 70^\circ\), \(\theta_2 = 30^\circ\). \(z_1 = 3\operatorname{cis}70^\circ\), \(z_2 = 3\operatorname{cis}30^\circ = \frac{3\sqrt3}{2} + \frac32 i\).

b) \(z_1 = \operatorname{cis}(-50^\circ)\), \(z_2 = \operatorname{cis}20^\circ\). \(\frac{z_1z_2}{z_3} = \operatorname{cis}(-40^\circ) \approx 0.766 - 0.643i\).

Problem 6 De Moivre's theorem

\([r\operatorname{cis}\theta]^n = r^n\operatorname{cis}n\theta\).

a) Calculate \((1 + i)^{12}\) and \((-1 + \sqrt3 i)^{9}\).

b) Calculate \(z^{-10}\) for \(z = 1 - i\).

c) Calculate \((\sqrt3 + i)^{30}\).

d) \(z = \frac{\sqrt3}{2} - \frac12 i\). Calculate \(z^{18}\).

Show solution

a) \((\sqrt2\operatorname{cis}45^\circ)^{12} = 64\operatorname{cis}540^\circ = -64\). \((2\operatorname{cis}120^\circ)^9 = 512\operatorname{cis}1080^\circ = 512\).

b) \(z = \sqrt2\operatorname{cis}(-45^\circ)\), \(z^{-10} = 2^{-5}\operatorname{cis}450^\circ = \frac{i}{32}\).

c) \((2\operatorname{cis}30^\circ)^{30} = 2^{30}\operatorname{cis}900^\circ = -2^{30}\).

d) \(z = \operatorname{cis}(-30^\circ)\), \(z^{18} = \operatorname{cis}(-540^\circ) = -1\).

Problem 7 Powers in the plane

a) \(z = \frac{\sqrt2}{2} + \frac{\sqrt2}{2}i\). Calculate \(z^2, z^3, \dots, z^8\) and show them in the plane.

b) Write \(\dfrac{\left(2\operatorname{cis}\frac{\pi}{6}\right)^3\left(\operatorname{cis}\frac{\pi}{4}\right)^2}{\left(4\operatorname{cis}\frac{\pi}{3}\right)^2}\) as \(a + bi\).

c) \(|u| = |v| = 1\), \(\operatorname{Arg}(uv) = 50^\circ\), \(\operatorname{Arg}\left(\frac uv\right) = 10^\circ\). Find \(u^9 + v^9\).

Show solution

a) \(z = \operatorname{cis}45^\circ\): \(z^k = \operatorname{cis}(45k)^\circ\): \(i\), \(\frac{-\sqrt2 + \sqrt2 i}{2}\), \(-1\), \(\frac{-\sqrt2 - \sqrt2 i}{2}\), \(-i\), \(\frac{\sqrt2 - \sqrt2 i}{2}\), \(1\). They are the vertices of a regular octagon on the unit circle.

b) \(\frac{8\operatorname{cis}\frac{\pi}{2}\cdot\operatorname{cis}\frac{\pi}{2}}{16\operatorname{cis}\frac{2\pi}{3}} = \frac12\operatorname{cis}\left(\pi - \frac{2\pi}{3}\right) = \frac12\operatorname{cis}\frac{\pi}{3} = \frac14 + \frac{\sqrt3}{4}i\).

c) \(\arg u = 30^\circ\), \(\arg v = 20^\circ\). \(u^9 = \operatorname{cis}270^\circ = -i\), \(v^9 = \operatorname{cis}180^\circ = -1\). Sum \(-1 - i\).