1.
[latex]\left(2, \dfrac{2\pi}{3}\right)[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Practice each skill in the Homework Problems listed.
For Problems 1–8, use the grid below to plot the points whose polar coordinates are given.
[latex]\left(2, \dfrac{2\pi}{3}\right)[/latex]
[latex](3, \dfrac{5\pi}{4})[/latex]
[latex]\left(3, \dfrac{3\pi}{2}\right)[/latex]
[latex](4, \pi)[/latex]
[latex]\left(-4, \dfrac{\pi}{6}\right)[/latex]
[latex]\left(-1, \dfrac{\pi}{2}\right)[/latex]
[latex]\left(-1, \dfrac{7\pi}{4}\right)[/latex]
[latex]\left(-2, \dfrac{5\pi}{3}\right)[/latex]
For Problems 9–16, give polar coordinates for each point shown below, with [latex]r \ge 0[/latex] and [latex]0 \le \theta \le 2\pi\text{.}[/latex]
[latex]A[/latex]
[latex]B[/latex]
[latex]C[/latex]
[latex]D[/latex]
[latex]E[/latex]
[latex]F[/latex]
[latex]G[/latex]
[latex]H[/latex]
For Problems 17–24, convert the polar coordinates to Cartesian coordinates.
[latex]\left(6, \dfrac{2\pi}{3}\right)[/latex]
[latex]\left(5, \dfrac{7\pi}{6}\right)[/latex]
[latex]\left(-3, \dfrac{3\pi}{4}\right)[/latex]
[latex]\left(-4, \dfrac{5\pi}{3}\right)[/latex]
[latex](2.4, 3.6)[/latex]
[latex](1.7, 5.2)[/latex]
[latex](-2, 1.5)[/latex]
[latex](-3, 4.5)[/latex]
For Problems 25–32, 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](7,7)[/latex]
[latex](-7,-7)[/latex]
[latex]\left(\sqrt{6}, -\sqrt{2}\right)[/latex]
[latex](-\sqrt{3}, 3)[/latex]
[latex](-3, -2)[/latex]
[latex](1, 4)[/latex]
[latex](-2,0)[/latex]
[latex](0, -5)[/latex]
In Problems 33–38, polar coordinates are given.
[latex]\left(2, \dfrac{5\pi}{6}\right)[/latex]
[latex]\left(1, \dfrac{4\pi}{3}\right)[/latex]
[latex](3,\pi)[/latex]
[latex](4,0)[/latex]
[latex](2.3, 5.2)[/latex]
[latex](1.2, 1.2)[/latex]
For Problems 39–44, sketch the region described by the inequalities.
[latex]r \le 3[/latex]
[latex]2 \le r \le 3[/latex]
[latex]\dfrac{\pi}{6} \le \theta \le \dfrac{\pi}{3}[/latex]
[latex]0 \le \theta \le \dfrac{\pi}{2}[/latex]
[latex]r \ge 4,~ \dfrac{\pi}{2} \le \theta \le \dfrac{3\pi}{4}[/latex]
[latex]2 \le r \le 3,~ \dfrac{5\pi}{3} \le \theta \le \dfrac{11\pi}{6}[/latex]
For Problems 45–50, write inequalities to describe the region.
For Problems 51–64, convert the equation into Cartesian coordinates.
[latex]r^2 = 2[/latex]
[latex]r = 2[/latex]
[latex]r = 4\cos \theta[/latex]
[latex]r = \sin \theta[/latex]
[latex]r = \csc \theta[/latex]
[latex]r = 2\sec \theta[/latex]
[latex]\tan \theta = 2[/latex]
[latex]r^2 = \tan \theta[/latex]
[latex]r\sec \theta = 3[/latex]
[latex]\theta = \dfrac{\pi}{3}[/latex]
[latex]r = \dfrac{2}{1+\sin\theta}[/latex]
[latex]r = \dfrac{4}{4-\cos\theta}[/latex]
[latex]2r\cos \theta + r\sin \theta = 1[/latex]
[latex]\tan \theta = r\cos\theta -2[/latex]
For Problems 65–72, convert the equation into polar coordinates.
[latex]x = 2[/latex]
[latex]y^2 = 2x-x^2[/latex]
[latex]2xy=1[/latex]
[latex]y=-x[/latex]
[latex]y^2=4x[/latex]
[latex]x^2-y^2=1[/latex]
[latex]x^2+y^2=4\sqrt{x^2+y^2}[/latex]
[latex](x^2+y^2)^3 = 9x^2y^2[/latex]
Use the law of cosines to prove the distance formula in polar coordinates:
[latex]d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos (\theta_2 - \theta_1)}[/latex]
Show that the graph of [latex]r=a\cos \theta + b\sin \theta[/latex] is a circle. Find its center and radius.