8.E: Exercises

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Sep 17, 2022
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- 8.2: Spherical and Cylindrical Coordinates
- 9: Vector Spaces

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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.

$z=x^2+y^2$
$x^2-y^2=1$
$z^2+x^2+y^2=6$
$z=\sqrt{x^2+y^2}$
$y=x$
$z=x$

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Exercise 8.E.1\PageIndex{1}

Exercise \PageIndex{2}\PageIndex{2}

Exercise \PageIndex{3}\PageIndex{3}

Exercise \PageIndex{4}\PageIndex{4}

Exercise \PageIndex{5}\PageIndex{5}

Exercise \PageIndex{6}\PageIndex{6}

Exercise \PageIndex{7}\PageIndex{7}

Exercise \PageIndex{8}\PageIndex{8}

Exercise \PageIndex{9}\PageIndex{9}

Exercise \PageIndex{10}\PageIndex{10}

Exercise \PageIndex{11}\PageIndex{11}

Exercise \PageIndex{12}\PageIndex{12}

Exercise \PageIndex{13}\PageIndex{13}

Exercise \PageIndex{14}\PageIndex{14}

Exercise \PageIndex{15}\PageIndex{15}

Exercise \PageIndex{16}\PageIndex{16}

Exercise \PageIndex{17}\PageIndex{17}

Exercise \PageIndex{18}\PageIndex{18}

Exercise \PageIndex{19}\PageIndex{19}

Exercise \PageIndex{20}\PageIndex{20}

Exercise \PageIndex{21}\PageIndex{21}

Exercise \PageIndex{22}\PageIndex{22}

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