1.
[latex]\left(4, \dfrac{5\pi}{6}\right)[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Problems: #4, 8, 12, 14, 20, 28, 30, 36, 38, 46, 50, 52, 53, 58, 66, 68, 72, 76, 82
For Problems 1–4, use the grid at right to plot the points whose polar coordinates are given.
[latex]\left(4, \dfrac{5\pi}{6}\right)[/latex]
[latex]\left(1, \dfrac{-5\pi}{3}\right)[/latex]
[latex]\left(-3, \dfrac{\pi}{2}\right)[/latex]
[latex]\left(2, \dfrac{7\pi}{4}\right)[/latex]
For Problems 5–8, convert the polar coordinates to Cartesian coordinates.
[latex]\left(1, \dfrac{5\pi}{4}\right)[/latex]
[latex]\left(0, \dfrac{\pi}{12}\right)[/latex]
[latex](3.4, -1.5)[/latex]
[latex](-5.6, -1.1)[/latex]
For Problems 9–12, convert the Cartesian coordinates to polar coordinates with [latex]r \ge 0[/latex] and [latex]0 \le \theta \le 2\pi\text{.}[/latex] Give exact values for [latex]r[/latex] and [latex]\theta\text{.}[/latex]
[latex](-3, 3)[/latex]
[latex](0, -2)[/latex]
[latex](5, -2)[/latex]
[latex](-15, -8)[/latex]
For Problems 13–16, sketch the region described by the inequalities.
[latex]r \ge 0,~ \dfrac{-\pi}{4} \le \theta \le \dfrac{\pi}{4}[/latex]
[latex]1 \le r \le 3,~0 \le \theta \le \pi[/latex]
[latex]0 \le r \le 2[/latex]
[latex]r \ge 4[/latex]
For Problems 17–20, convert the equation into Cartesian coordinates.
[latex]r=1[/latex]
[latex]r=-3\sec \theta[/latex]
[latex]r=\dfrac{6}{1-2\cos\theta}[/latex]
[latex]3\tan \theta = 6r\sin \theta - 1[/latex]
For Problems 21–24, convert the equation into polar coordinates.
[latex]x+y = 2[/latex]
[latex]\sqrt{x^2+y^2} = 4y[/latex]
[latex]\dfrac{y}{x}=\sqrt{x^2+y^2}[/latex]
[latex]y^2 = 2y = x - x^2[/latex]
For Problems 25–28, use the catalog of polar graphs to help you identify and sketch the curve. Check your work by graphing with a calculator.
[latex]r=3[/latex]
[latex]\theta = \dfrac{3\pi}{4}[/latex]
[latex]r=6\cos \theta[/latex]
[latex]r^2=9\sin 2\theta[/latex]
For Problems 29–32, write a polar equation for the graph.
For Problems 33–36, find the coordinates of the intersection points of the two curves analytically. Then graph the curves to verify your answers.
[latex]r=3+2\sin \theta,~r=4[/latex]
[latex]r=3\cos\theta,~r=\sqrt{3}\sin \theta[/latex]
[latex]r=4\sin \theta,~r=-4\cos \theta[/latex]
[latex]r=2+6\sin \theta,~r=4\sin \theta[/latex]
For Problems 37–40, perform the indicated operations on the complex numbers.
[latex]\dfrac{5-10i}{2-i}[/latex]
[latex](4-7i)(1+i)[/latex]
[latex]5i(2-i)-(7+6i)[/latex]
[latex]-8+3i+\dfrac{9-4i}{i}[/latex]
For Problems 41–44, evaluate the polynomial for the given values of the variable.
[latex]z^2+4z+6[/latex]
[latex]z^2-6z+12[/latex]
[latex]3w^2-18w+31[/latex]
[latex]2w^2+8w+11[/latex]
Verify that [latex]z_1=2+i[/latex] and [latex]z_2=2-i[/latex] are roots of the equation [latex]x^2-4x+5=0\text{.}[/latex]
Verify that [latex]z_1=-3+4i[/latex] and [latex]z_2=-3-4i[/latex] are roots of the equation [latex]x^2+6x+25=0\text{.}[/latex]
For Problems 47–50, expand the product of polynomials.
[latex][z-(-2+i)][z-(-2-i)][/latex]
[latex][w-(1+3i)][w-(1-3i)])][/latex]
[latex][s+(5+4i)][s+(5-4i)][/latex]
[latex][x+(-6+i)][x+(-6-i)][/latex]
For Problems 51–52, sketch the set of points in the complex plane.
[latex]z_1=-3+2i,~[/latex] [latex]z_2=-3-2i,~[/latex] [latex]z_3=3+2i,~[/latex] [latex]z_4=3-2i[/latex]
[latex]w_1=4-6i,~[/latex] [latex]w_2=4+6i,~[/latex] [latex]w_3=-4-6i,~[/latex] [latex]w_4=-4+6i[/latex]
For Problems 53–56,
[latex]-1+7i[/latex]
[latex]2-5i[/latex]
[latex]3-\sqrt{2}i[/latex]
[latex]4+\sqrt{3}i[/latex]
For Problems 57–60, write the complex numbers in standard form. Give exact values for your answers.
[latex]10(\cos \dfrac{-\pi}{6} + i\sin \dfrac{-\pi}{6})[/latex]
[latex]8(\cos \dfrac{5\pi}{4} + i\sin \dfrac{5\pi}{4})[/latex]
[latex]5\sqrt{2}(\cos \dfrac{\pi}{4} + i\sin \dfrac{\pi}{4})[/latex]
[latex]6\sqrt{3}(\cos \dfrac{-\pi}{3} + i\sin \dfrac{-\pi}{3})[/latex]
For Problems 61–66, write the complex numbers in polar form. Give exact values for your answers.
[latex]3-3i[/latex]
[latex]-4-4i[/latex]
[latex]-5[/latex]
[latex]-7i[/latex]
[latex]-1-\sqrt{3}i[/latex]
[latex]6+2\sqrt{3}i[/latex]
For Problems 67–70, find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]
[latex]z_1=8\left(\cos \dfrac{\pi}{6} + i\sin \dfrac{\pi}{6}\right)[/latex] [latex]z_2=2\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]
[latex]z_1=9\left(\cos \dfrac{-2\pi}{3} + i\sin \dfrac{-2\pi}{3}\right)[/latex][latex]z_2=3\left(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}\right)[/latex]
[latex]z_1=5\left(\cos \dfrac{-7\pi}{12} + i\sin \dfrac{-7\pi}{12}\right)[/latex][latex]z_2=\dfrac{1}{2}\left(\cos \dfrac{\pi}{4} + i\sin \dfrac{\pi}{4}\right)[/latex]
[latex]z_1=14\left(\cos \dfrac{3\pi}{2} + i\sin \dfrac{3\pi}{2}\right)[/latex][latex]z_2=2\left(\cos \dfrac{\pi}{6} + i\sin \dfrac{\pi}{6}\right)[/latex]
For Problems 71–74, find the power.
[latex]\left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\right)^{12}[/latex]
[latex]\left(1-\sqrt{3}i\right)^6[/latex]
[latex]\left(-\sqrt{5} + \sqrt{5}i\right)^{-4}[/latex]
[latex](-1-i)^{-8}[/latex]
For Problems 75–78,
The square roots of [latex]-16i[/latex]
The cube roots of [latex]-8[/latex]
The cube roots of [latex]-27i[/latex]
The square roots of [latex]-2+2\sqrt{3}i[/latex]
For Problems 79–82, solve the equation.
[latex]z^6+27=0[/latex]
[latex]z^4-6z^2+12=0[/latex]
[latex]z^4+2z^2+4=0[/latex]
[latex]z^6-8=0[/latex]