Parametric Equations, Polar Coordinates and Conic Sections

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(3,4\right)$

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(6,2\sqrt{3}\right)$

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(-2,0\right)$

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.

$\left(3,-3\right)$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$\theta =\mathrm{\pi }$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$\theta =\frac{\mathrm{\pi }}{4}$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$\theta =-\frac{\mathrm{\pi }}{6}$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$\theta =\frac{7\mathrm{\pi }}{6}$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=2 \cos \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates. 

$r=5 \sin \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=-3 \sec \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$r=6 \csc \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\tan \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\sin 2\theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r^2=\sin \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r^2=\cos \theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\sin 4\theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r=\cos 4\theta$

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r={\sin }^3\theta$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$x=0$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=0$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$x=4$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=x$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=\sqrt{3}x$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=-3$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=x+1$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y=mx$

Convert the equations given in polar coordinates to equations in rectangular coordinates. 

$r=1,r=-2,r=3$

Convert the equations given in polar coordinates to equations in rectangular coordinates.  

$r=k$ for each positive integer k less than 10. 

convert the equations given in polar coordinates to equations in rectangular coordinates 

$\theta =0,\theta =\frac{\mathrm{\pi }}{4},\theta =-\frac{\mathrm{\pi }}{2}$

Convert the equations given in polar coordinates to equations in rectangular coordinates  

$\theta = \frac{k\mathrm{\pi }}{6}$, for each positive integer k less than 12. 

Ian has one cam that is 1 inch wide, with a point where

the rope attaches on its right corner, so that the device

pivots on its left corner. In other words, the left corner

of the cam becomes the center of a circle, and the point

where the rope attaches follow the curve 

(a) If a crack is wide, at what angle must Ian

turn the device in order to put it into the crack?

(Hint: compute the x-coordinate of the right edge of the

device.)

(b) If Ian falls, the rope will pull on the right edge of the

cam while the left edge remains wedged in a fixed

position. What happens to the width of the cam in

the crack?

61. Ian has another camming device, one that has two arms

extending from a point where the rope attaches in the

center.

The outside edge of each arm follows the curve $r=0.75$.

The ends of the arms make an angle $\pm \theta$ with the rope,

and the rope pulls along the ray$\theta =0$. Ian can retract the

arms so that θ is small in order to put the device into a

crack, and then a spring pulls them back so that each arm

wedges against the walls of the crack.

(a) Ian puts the device into a horizontal crack that is

$1$ inch tall. What angle do the arms make with the

rope? (Hint: Compute the $y$-coordinates of the two arm

ends.)

(b) What happens if Ian falls, causing the rope to pull

outwards (rightwards) in the crack?

Prove that the graph of the equation:  $r=ksec\theta$, $-\frac{\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$ is a vertical line for any value of  $k\ne 0$

. Prove that the graph of the equation $r=kcoesc\theta$, $0<\theta <\pi$, is a horizontal line for any value of $k\ne 0$ 

Let $a\ne 0$. Prove that the graph of the equation $r = \frac{a}{1 - \cos \theta }$ is a parabola in a polar coordinate system. When $a\ne 0$, what is the graph of the equation $r=\frac{a}{1-\sin \theta }$ ? 

Let $a\ne 0$ and $0<b<1$ Prove that the graph of the equation $r=\frac{a}{1-b\cos \theta }$ is an ellipse in a polar coordinate system. When and b > 1, what is the graph of the equation $r = \frac{a}{1-b\sin \theta }$?

Show that the graph of the equation $r=k\cos \theta$ is a circle tangent to the y-axis for any $k\ne 0$. What are the center and radius of the circle? 

Finish the proof of Theorem $9.8$ by showing that the graph of the equation $r=2k\sin \theta$ is a circle with center $\left(0,k\right)$ and radius $k$ tangent to the x-axis for any $k\ne 0$. 

Modify the proof of Theorem $9.8$ to show that the graph of the equation $r=k\sin \theta +l\cos \theta$ is a circle. Find the center and radius in terms of $k$  and $l$ . 

Find and prove a formula for the distance between the

points $(r_1,{\theta }_1)$ and $(r_2,{\theta }_1)$ when they are plotted in a polar

coordinate system.

Finding points of intersection: Graph the function $y=1+\sin x and y=\cos x$ and find all the points of intersection

Finding points of intersection: Graph the function $y=\frac{1}{2}+\cos x and y=\frac{1}{2}-\cos x$ find all points of intersection

Translation of graphs: what is the relationship between graphs of the function $$\begin{gathered} y=f\left(x\right) and y=f(x-h) \\ where k>0 \end{gathered}$$

Translation of graphs: what is the relationship between the functions $y=f\left(x\right) and y=f\left(x\right)+k$ where k > 0

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If π 2 < θ < π, then the point (r, θ) is located in the second quadrant when it is plotted in a polar coordinate system.

(b) True or False: The graph of r = sin 5θ is a five-petaled rose.

(c) True or False: The graph of r = cos 6θ is a six-petaled rose.

(d) True or False: If a graph in the polar plane is symmetrical with respect to the origin, then for every polar point (r, θ) on the graph, the polar point (−r, θ +2π)

is also on the graph.

(e) True or False: The graph of a polar function r = f (θ) is symmetrical with respect to the y-axis if, for every point (r, θ) on the graph, the point (r,−θ) is also on the graph.

(f) True or False: When k is a positive integer, the polar roses r = sin kθ and r = cos kθ are symmetrical with respect to both the x-axis and y-axis if and only if k is even.

(g) True or False: In the rectangular coordinate system the graph of the equation (x 2+y2) 2 = k(x 2−y2) is a lemniscate for every k > 0.

(h) True or False: In the rectangular coordinate system the only function y = f (x) that is symmetrical with respect to both the y-axis and the origin is y = 0.

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An equation in polar coordinates whose graph is a cardioid.

(b) An equation in polar coordinates whose graph is a limacon. 

(c) An equation in polar coordinates whose graph is a lemniscate. 

How can the graph of an equation $r=f\left(\theta \right)$ in the $\theta -r$-plane be used to provide information about the graph of the same equation in the polar plane? 

Explain the significance of the following figure: 

What is being shown in this figure?