Skip to main content
Mathematics LibreTexts

8.E: Exercises

  • Page ID
    14547
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Exercise \(\PageIndex{1}\)

    In the following, polar coordinates \((r,θ)\) for a point in the plane are given. Find the corresponding Cartesian coordinates.

    1. \((2,\pi /4)\)
    2. \((-2, \pi/4)\)
    3. \((3, \pi/3)\)
    4. \((-3, \pi/3)\)
    5. \((2,5\pi /6)\)
    6. \((-2, 11\pi /6)\)
    7. \((2,\pi /2)\)
    8. \((1,3\pi /2)\)
    9. \((-3, 3\pi /4)\)
    10. \((3, 5\pi /4)\)
    11. \((-2, \pi /6)\)

    Exercise \(\PageIndex{2}\)

    Consider the following Cartesian coordinates \((x, y)\). Find polar coordinates corresponding to these points.

    1. \((-1,1)\)
    2. \((\sqrt{3},-1)\)
    3. \((0,2)\)
    4. \((-5,0)\)
    5. \((-2\sqrt{3},2)\)
    6. \((2,-2)\)
    7. \((-1,\sqrt{3})\)
    8. \((-1,-\sqrt{3})\)

    Exercise \(\PageIndex{3}\)

    The following relations are written in terms of Cartesian coordinates \((x, y)\). Rewrite them in terms of polar coordinates, \((r,\theta )\).

    1. \(y=x^2\)
    2. \(y=2x+6\)
    3. \(x^2+y^2=4\)
    4. \(x^2-y^2=1\)

    Exercise \(\PageIndex{4}\)

    Use a calculator or computer algebra system to graph the following polar relations.

    1. \(r=1-\sin (2\theta ),\:\theta\in [0,2\pi ]\)
    2. \(r=\sin (4\theta ),\:\theta\in [0,2\pi ]\)
    3. \(r=\cos (3\theta )+\sin (2\theta ),\: \theta\in [0,2\pi]\)
    4. \(r=\theta,\:\theta\in [0,15]\)

    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.

    1. \((5,\frac{5\pi}{6},-3)\)
    2. \((3,\frac{\pi}{3},4)\)
    3. \((4,\frac{2\pi}{3},1)\)
    4. \((2,\frac{3\pi}{4},-2)\)
    5. \((3,\frac{3\pi}{2},-1)\)
    6. \((8,\frac{11\pi}{6},-11)\)

    Exercise \(\PageIndex{14}\)

    The following are the Cartesian coordinates of points, \((x, y,z)\). Find the cylindrical and spherical coordinates of these points.

    1. \((\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)\)
    2. \((\frac{3}{2},\frac{3}{2}\sqrt{3},2)\)
    3. \((-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)\)
    4. \((-\frac{5}{2},\frac{5}{2}\sqrt{3},23)\)
    5. \((-\sqrt{3},-1,-5)\)
    6. \((\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)\)
    7. \((\sqrt{2},\sqrt{6},2\sqrt{2})\)
    8. \((-\frac{1}{2}\sqrt{3},\frac{3}{2},1)\)
    9. \((-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})\)
    10. \((-\sqrt{3}1,2\sqrt{3})\)
    11. \((-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})\)

    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.

    1. \((4,\frac{\pi}{4},\frac{5\pi}{6})\)
    2. \((2,\frac{\pi}{3},\frac{2\pi}{3})\)
    3. \((3,\frac{5\pi}{6},\frac{3\pi}{2})\)
    4. \((4,\frac{\pi}{2},\frac{7\pi}{4})\)
    5. \((4,\frac{2\pi}{3},\frac{\pi}{6})\)
    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.

    1. \(z=x^2+y^2\)
    2. \(x^2-y^2=1\)
    3. \(z^2+x^2+y^2=6\)
    4. \(z=\sqrt{x^2+y^2}\)
    5. \(y=x\)
    6. \(z=x\)

    Exercise \(\PageIndex{22}\)

    The following are described in Cartesian coordinates. Rewrite them in terms of cylindrical coordinates.

    1. \(z=x^2+y^2\)
    2. \(x^2-y^2=1\)
    3. \(z^2+x^2+y^2=6\)
    4. \(z=\sqrt{x^2+y^2}\)
    5. \(y=x\)
    6. \(z=x\)

    This page titled 8.E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?