Polar Coordinates and Equations in Polar Form: Problems with Solutions

Solution:
We use the transformation equations

$x=r\cos\theta$
$y=r\sin \theta $
where $r=0$   $\theta =\frac{\pi}{2}$

$x=0\cdot \cos(\frac{\pi}{2})=0$

$y=0\cdot \sin (\frac{\pi}{2})=0$

So $(0,\frac{\pi}{2})$ is equivalent to $(0,0)$ in Cartesian Coordinates.

Solution:
Answer: $(-\sqrt{2},\frac{\pi}{4})\equiv (-1,-1)$

We use the transformation equations
$x=r\cos\theta$
$y=r\sin\theta $   where $r=-\sqrt{2}\qquad \theta =\pi /4$

$x=-\sqrt{2}\cdot \cos(\pi /4)=-\sqrt{2} \times \frac{\sqrt{2}}{2}=-1$
$y=-\sqrt{2}\cdot \sin (\pi /4)=-\sqrt{2} \times \frac{\sqrt{2}}{2}=-1$

The point in Cartesian coordinates is $(-1,-1)$.

Solution:
$y=10\Longrightarrow y=r\sin\theta =10\Longrightarrow r=\frac{10}{\sin\theta }=10\csc \theta $

Solution:
Answer: $r^{2}\left( \cos 2\theta \right) =4$

We use the transformation equations
$x=rcos\theta \qquad y=r\sin\theta$

$x^{2}-y^{2}=4\Longrightarrow \left( rcos\theta \right)^{2}-\left(r\sin\theta \right) ^{2}=4\Longrightarrow r^{2}\left( \cos ^{2}\theta -\sin^{2}\theta \right) =4$

$r^{2}\left( \cos 2\theta \right) =4$
This is the equation of a hyperbola, and here is its graph.

Solution:
We use the transformation equations
$x=r\cos\theta \qquad y=r\sin\theta $

$y^{2}=4x$ $\Longrightarrow \left( r\sin\theta \right) ^{2}=4r\cos\theta \Longrightarrow r^{2}\sin^{2}\theta =4r\cos\theta \Longrightarrow r=4\frac{\cos \theta }{sin^{2}\theta }$

This is the equation of a parabola.

Solution:
$\theta =2\frac{\pi }{3};r\leq -2$

This is the graph of a ray that makes an angle of $2\frac{\pi}{3}$ with the positive $x$-axis, but goes in the opposite direction starting at $-2.$

Solution:
Answer: $0\leq \theta \leq \pi \qquad r=-1$

This is half a circle that starts at $\pi$ and goes to $2\pi $.
I know that the interval starts at $0$, but $r$ is negative.

Solution:
Answer: $-\frac{\pi }{4}\leq \theta \leq \frac{\pi }{4}\qquad -1\leq r\leq 1$

The graph of this set of inequalities is two wedges cut out of the circle with radius of $-1$ and $1$, and all points in blue that are between those two values by the lines $\pm \frac{\pi }{4}$.

Solution:
Answer: $y=4$

$r\sin\theta =4\Longrightarrow y=4$

Since $y=r\sin \theta $ is the transformation equation.

This is an equation of a horizontal line through point $(0,4)$.

Solution:
Answer: $y=x+4$

We use the transformation equations
$x=r\cos\theta \qquad y=r\sin\theta$

So $r\sin\theta =r\cos\theta +4\Longrightarrow y=x+4$

This is a straight line with slope $1$ and $y$-intercept at $(0,4)$

Solution:
We match both equations
$r=\sin \theta $ and $r=\sin 2\theta $

then $\sin \theta =\sin 2\theta =2\sin \theta \cos \theta \Longrightarrow 2\cos \theta =1\Longrightarrow \cos \theta =\frac{1}{2}$

So, $\theta =\arccos \frac{1}{2}= \frac{1}{3}\pi ,\frac{5}{3}\pi $

when $\theta =\frac{1}{3}\pi \Longrightarrow $ $r=\sin \left( \frac{1}{3}\pi \right) =\frac{\sqrt{3}}{2}$ and $r=\sin \left( \frac{2}{3}\pi \right) =\frac{\sqrt{3}}{2}$

and when $\theta =\frac{5}{3}\pi \Longrightarrow r=\sin \left( \frac{5}{3}\pi \right) =-\frac{\sqrt{3}}{2}$ and $r=\sin \left( \frac{10}{3}\pi \right) = -\frac{\sqrt{3}}{2}$

The points of intersection are $\left( \frac{\sqrt{3}}{2},\frac{1}{3}\pi \right) ,\left( -\frac{\sqrt{3}}{2},\frac{5}{3}\pi \right) $

Solution:
Answer: Hyperbola, ellipse.

$r=\frac{2}{1-2\sin \theta }$

If we compare each of the terms in the given equation in polar form $r=\frac{ep}{1-e\sin\theta }$
we can see that $e=2$.

Therefore, the conic section is a hyperbola.

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$r=\frac{3}{4+\cos \theta }=\frac{3/4}{1+1/4\cos \theta }$
and we compare with $r=\frac{ep}{1+e\sin\theta }$ then $e=\frac{1}{4}$

So, the conic section is an ellipse.

Solution:
$r=\frac{4}{3-2\sin \theta }=\frac{4/3}{1-2/3\sin \theta }$
We see that $e=\frac{2}{3}$ then
The equation $r=\frac{4}{3-2\sin \theta}$ represents an ellipse.

Solution:
The conic section represents a parabola which axis is horizontal, since $r$ is not defined when $\theta=0$

The vertex of the parabola is $\theta =\pi $
Vertex: $(\frac{1}{2},\pi )$

$y-axis$ intersection: $(1,\frac{\pi }{2});$ $(1,\frac{3\pi }{2})$

Solution:
We see that $e=2$.
So the equation $r=\frac{2}{1+2\cos \theta }$ represents a hyperbola,
which axis is horizontal, along the x-axis. The vertices, which are the edges of the transverse axis of the hyperbola, are $\theta =0$ and $\theta =\pi $

Vertices: $(\frac{2}{3},0);\ \ (-2,\pi )$

$y-axis$ intersection at: $(2,\frac{\pi }{2});\ \ (2,\frac{3\pi }{2})$