8.E: Exercises
-
- Last updated
- Save as PDF
Exercise \(\PageIndex{1}\)
In the following, polar coordinates \((r,θ)\) for a point in the plane are given. Find the corresponding Cartesian coordinates.
- \((2,\pi /4)\)
- \((-2, \pi/4)\)
- \((3, \pi/3)\)
- \((-3, \pi/3)\)
- \((2,5\pi /6)\)
- \((-2, 11\pi /6)\)
- \((2,\pi /2)\)
- \((1,3\pi /2)\)
- \((-3, 3\pi /4)\)
- \((3, 5\pi /4)\)
- \((-2, \pi /6)\)
Exercise \(\PageIndex{2}\)
Consider the following Cartesian coordinates \((x, y)\). Find polar coordinates corresponding to these points.
- \((-1,1)\)
- \((\sqrt{3},-1)\)
- \((0,2)\)
- \((-5,0)\)
- \((-2\sqrt{3},2)\)
- \((2,-2)\)
- \((-1,\sqrt{3})\)
- \((-1,-\sqrt{3})\)
Exercise \(\PageIndex{3}\)
The following relations are written in terms of Cartesian coordinates \((x, y)\). Rewrite them in terms of polar coordinates, \((r,\theta )\).
- \(y=x^2\)
- \(y=2x+6\)
- \(x^2+y^2=4\)
- \(x^2-y^2=1\)
Exercise \(\PageIndex{4}\)
Use a calculator or computer algebra system to graph the following polar relations.
- \(r=1-\sin (2\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\sin (4\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\cos (3\theta )+\sin (2\theta ),\: \theta\in [0,2\pi]\)
- \(r=\theta,\:\theta\in [0,15]\)
Exercise \(\PageIndex{5}\)
Graph the polar equation \(r = 1+\sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{6}\)
Graph the polar equation \(r = 2+\sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{7}\)
Graph the polar equation \(r = 1+2 \sinθ\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{8}\)
Graph the polar equation \(r = 2+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{9}\)
Graph the polar equation \(r = 1+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{10}\)
Graph the polar equation \(r = 1+\sin(3θ)\) for \(θ ∈ [0, 2π]\).
Exercise \(\PageIndex{11}\)
Describe how to solve for \(r\) and \(θ\) in terms of \(x\) and \(y\) in polar coordinates.
Exercise \(\PageIndex{12}\)
This problem deals with parabolas, ellipses, and hyperbolas and their equations. Let \(l\), \(e > 0\) and consider \[r=\frac{l}{1\pm e\cos\theta}\nonumber\] Show that if \(e = 0\), the graph of this equation gives a circle. Show that if \(0 < e < 1\), the graph is an ellipse, if \(e = 1\) it is a parabola and if \(e > 1\), it is a hyperbola.
Exercise \(\PageIndex{13}\)
The following are the cylindrical coordinates of points, \((r,θ,z)\). Find the Cartesian and spherical coordinates of each point.
- \((5,\frac{5\pi}{6},-3)\)
- \((3,\frac{\pi}{3},4)\)
- \((4,\frac{2\pi}{3},1)\)
- \((2,\frac{3\pi}{4},-2)\)
- \((3,\frac{3\pi}{2},-1)\)
- \((8,\frac{11\pi}{6},-11)\)
Exercise \(\PageIndex{14}\)
The following are the Cartesian coordinates of points, \((x, y,z)\). Find the cylindrical and spherical coordinates of these points.
- \((\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)\)
- \((\frac{3}{2},\frac{3}{2}\sqrt{3},2)\)
- \((-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)\)
- \((-\frac{5}{2},\frac{5}{2}\sqrt{3},23)\)
- \((-\sqrt{3},-1,-5)\)
- \((\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)\)
- \((\sqrt{2},\sqrt{6},2\sqrt{2})\)
- \((-\frac{1}{2}\sqrt{3},\frac{3}{2},1)\)
- \((-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})\)
- \((-\sqrt{3}1,2\sqrt{3})\)
- \((-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})\)
Exercise \(\PageIndex{15}\)
The following are spherical coordinates of points in the form \((ρ,φ,θ)\). Find the Cartesian and cylindrical coordinates of each point.
- \((4,\frac{\pi}{4},\frac{5\pi}{6})\)
- \((2,\frac{\pi}{3},\frac{2\pi}{3})\)
- \((3,\frac{5\pi}{6},\frac{3\pi}{2})\)
- \((4,\frac{\pi}{2},\frac{7\pi}{4})\)
- \((4,\frac{2\pi}{3},\frac{\pi}{6})\)
- \((4,\frac{3\pi}{4},\frac{5\pi}{3})\)
Exercise \(\PageIndex{16}\)
Describe the surface \(φ = π/4\) in Cartesian coordinates, where \(φ\) is the polar angle in spherical coordinates.
Exercise \(\PageIndex{17}\)
Describe the surface \(θ = π/4\) in spherical coordinates, where \(θ\) is the angle measured from the positive \(x\) axis.
Exercise \(\PageIndex{18}\)
Describe the surface \(r=5\) in Cartesian coordinates, where \(r\) is one of the cylindrical coordinates.
Exercise \(\PageIndex{19}\)
Describe the surface \(\rho =4\) in Cartesian coordinates, where \(\rho\) is the distance to the origin.
Exercise \(\PageIndex{20}\)
Give the cone described by \(z=\sqrt{x^2+y^2}\) in cylindrical coordinates and in spherical coordinates.
Exercise \(\PageIndex{21}\)
The following are described in Cartesian coordinates. Rewrite them in terms of spherical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)
Exercise \(\PageIndex{22}\)
The following are described in Cartesian coordinates. Rewrite them in terms of cylindrical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)