# 8.E: Exercises

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

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

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

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

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

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

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

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

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

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

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

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

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

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

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

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## 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) .