1.
[latex]1,~i,~i^2,~i^3[/latex] and [latex]i^4[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Practice each skill in the Homework Problems listed.
For Problems 1–4, simplify and plot each complex number as a point on the complex plane.
[latex]1,~i,~i^2,~i^3[/latex] and [latex]i^4[/latex]
[latex]-1,~-i,~-i^2,~-i^3[/latex] and [latex]-i^4[/latex]
[latex]1+2i[/latex] and [latex]i(1+2i)[/latex]
[latex]3-4i[/latex] and [latex]i(3-4i)[/latex]
For Problems 5–8, write the complex numbers in standard form. Give exact values for your answers.
[latex]6\left(\cos \dfrac{2\pi}{3} + i\sin \dfrac{2\pi}{3}\right)[/latex]
[latex]4\left(\cos \dfrac{7\pi}{4} + i\sin \dfrac{7\pi}{4}\right)[/latex]
[latex]\sqrt{2}\left(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4}\right)[/latex]
[latex]\dfrac{3}{2}\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]
For Problems 9–12, write the complex numbers in standard form. Round your answers to hundredths.
[latex]5(\cos 5.2 + i\sin 5.2)[/latex]
[latex]3(\cos 3.5 + i\sin 3.5)[/latex]
[latex]12(\cos 115° + i\sin 115°)[/latex]
[latex]20(\cos 250° + i\sin 250°)[/latex]
For Problems 13–16, write the complex numbers in polar form. Give exact values for your answers.
[latex]3i[/latex] and [latex]-3i[/latex]
[latex]2+2i[/latex] and [latex]2-2i[/latex]
[latex]-3-\sqrt{3}i[/latex] and [latex]3-\sqrt{3}i[/latex]
[latex]2\sqrt{3}+2i[/latex] and [latex]-2\sqrt{3}+2i[/latex]
For Problems 17–22, write the complex numbers in polar form. Round your answers to hundredths.
[latex]-4+2i[/latex] and [latex]4-2i[/latex]
[latex]-3-8i[/latex] and [latex]3+8i[/latex]
[latex]9-5i[/latex] and [latex]9+5i[/latex]
[latex]2+6i[/latex] and [latex]2-6i[/latex]
[latex]3+4i,~ 3-4i,~ -3+4i,[/latex] and [latex]-3-4i[/latex]
[latex]1+3i,~ 1-3i,~ -1+3i,[/latex] and [latex]-1-3i[/latex]
What can you conclude about the polar forms of [latex]z[/latex] and [latex]\bar{z}\text{?}[/latex]
What can you conclude about the polar forms of [latex]z[/latex] and [latex]-z\text{?}[/latex]
For Problems 25–28, find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]
[latex]z_1=4\left(\cos \dfrac{4\pi}{3} + i\sin \dfrac{4\pi}{3}\right)[/latex] [latex]z_2=\dfrac{1}{2}\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]
[latex]z_1=6\left(\cos \dfrac{5\pi}{8} + i\sin \dfrac{5\pi}{8}\right)[/latex] [latex]z_2=\dfrac{3}{2}\left(\cos \dfrac{\pi}{8} + i\sin \dfrac{\pi}{8}\right)[/latex]
[latex]z_1=3\left(\cos \dfrac{3\pi}{5} + i\sin \dfrac{3\pi}{5}\right)[/latex] [latex]z_2=2\left(\cos \dfrac{3\pi}{10} + i\sin \dfrac{3\pi}{10}\right)[/latex]
[latex]z_1=4\left(\cos \dfrac{5\pi}{12} + i\sin \dfrac{5\pi}{12}\right)[/latex] [latex]z_2=6\left(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4}\right)[/latex]
For Problems 29–32, convert the complex number to polar form, then find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]
[latex]z_1=2i,~z_2=4i[/latex]
[latex]z_1=-2i,~z_2=3i[/latex]
[latex]z_1=2\sqrt{3}-2i\text{,}[/latex] [latex]~z_2=-1+i[/latex]
[latex]z_1=\sqrt{3}+i\text{,}[/latex] [latex]~z_2=-1+\sqrt{3}[/latex]
For Problems 33–38, find the power.
[latex](2+2i)^5[/latex]
[latex](\sqrt{2}-\sqrt{2}i)^6[/latex]
[latex](-1+\sqrt{3}i)^8[/latex]
[latex](\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)^{12}[/latex]
[latex](\sqrt{3}-i)^{10}[/latex]
[latex](1-i)^{20}[/latex]
For Problems 39–42, use De Moivre’s theorem to find the reciprocal.
[latex]2-2i[/latex]
[latex]3+\sqrt{3}i[/latex]
[latex]-\sqrt{2}+\sqrt{6}i[/latex]
[latex]-1-i[/latex]
For Problems 43–48,
The square roots of [latex]9i\text{.}[/latex]
The fourth roots of [latex]-81\text{.}[/latex]
The fifth roots of [latex]32\text{.}[/latex]
The cube roots of [latex]i\text{.}[/latex]
The cube roots of [latex]4\sqrt{3}+4i\text{.}[/latex]
The square roots of [latex]-2+2\sqrt{3}i\text{.}[/latex]
Show that any complex number of the form [latex]z=\cos\theta + i\sin \theta[/latex] lies on the unit circle in the complex plane.
Show that if [latex]{|z|}=1\text{,}[/latex] then [latex]\dfrac{1}{z}=\bar{z}\text{.}[/latex]
If [latex]n[/latex] is a positive integer, define
[latex]\omega_k = \cos \dfrac{2\pi k}{n} + i\sin \dfrac{2\pi k}{n}[/latex]
for [latex]k = 0,~1,~2,~ \cdots,~ n-1.[/latex] Show that [latex](\omega_k)^n = 1\text{.}[/latex] (We call [latex]\omega_k[/latex] an [latex]n^{th}[/latex] root of unity.)
Let [latex]\omega = \cos \dfrac{2\pi}{n} + i\sin \dfrac{2\pi}{n}\text{,}[/latex] where [latex]n[/latex] is a positive integer. Show that the [latex]n[/latex] distinct [latex]n^{th}[/latex] roots of unity are [latex]\omega,~\omega^2,~\omega^3,~ \cdots,~\omega^{n-1}\text{.}[/latex]
For Problems 55-60, solve the equation.
[latex]z^4+4z^2+8=0[/latex]
[latex]z^6+4z^3+8=0[/latex]
[latex]z^6-8=0[/latex]
[latex]z^4-9i=0[/latex]
[latex]z^4+2z^2+4=0[/latex]
[latex]z^4-2z^2+4=0[/latex]
Problems 63 and 64 show that multiplication by [latex]i[/latex] results in a rotation of [latex]90°\text{.}[/latex]
Suppose that [latex]z=a+bi[/latex] and that the real numbers [latex]a[/latex] and [latex]b[/latex] are both nonzero.
Suppose that [latex]z=a+bi[/latex] and that [latex]a[/latex] and [latex]b[/latex] are both real numbers.
Prove the product rule by following the steps.
Let [latex]z_1=r(\cos \alpha +i \sin \alpha)[/latex] and [latex]z_2=R(\cos \beta +i \sin \beta)\text{.}[/latex] Prove the quotient rule as follows: Set [latex]w=\dfrac{r}{R}(\cos (\alpha - \beta) + i\sin (\alpha - \beta)),[/latex] and show that [latex]z_1=wz_2\text{.}[/latex]