Mathematics 150BL

Assignment 10 -- Polar Coordinates

So far, we have been representing graphs as collections of points $(x,y)$ on the rectangular coordinate system. The corresponding equations for these graphs have been in either rectangular or parametric form. Here we work with the coordinate system called the polar coordinate system.

To form the polar coordinate system in the plane, we fix a point $O$, called the origin, and construct from $O$ an initial ray called the polar axis. Then each point $P$ in the plane can be assigned polar coordinates $(r,\theta )$ as follows.

$r=$ directed distance from $O$ to $P$

$\theta =$ directed angle, counterclockwise from polar axis to segment $OP$


3b10__11.png

With rectangular coordinates, each point $(x,y)$ has a unique representation. This is not true with polar coordinates. For instance, the coordinates $(r,\theta )$ and $(r,2\pi +\theta )$ represent the same point. Also, because $r$ is a directed distance, the coordinates $(r,\theta )$ and $(-r,\theta +\pi )$ represent the same point.

To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive $x-$axis and the pole with the origin. Because $(x,y)$ lies on a circle of radius $r$, it follows that $r^{2}=x^{2}+y^{2}$. Moreover, for $r>0$, the definition of the trigonometric functions implies that MATH, MATH, and MATH. That is, the polar coordinates $(r,\theta ) $ of a point are related to the rectangular coordinates $(x,y)$ of the point as follows.

$x=r\cos \theta $, $y=r\sin \theta $, MATH, and $r^{2}=x^{2}+y^{2}$.

Example: Polar-to-Rectangular Conversion

For the point MATH, MATH and MATH. Thus the rectangular coordinates are $(x,y)=(-2,0)$.

Example: Rectangular-to-Polar Conversion

For the point $(x,y)=(-1,1)$, MATH. Thus, MATH (since the original point is in the second quadrant). $r=$ MATH. Thus, one set of polar coordinates is: MATH (recall that polar coordinates are not unique).

Problems

1. Convert the following rectangular coordinates to polar coordinates.

a. MATHb. $(0,2)$

2. Convert the following polar coordinates to rectangular coordinates.

a. MATHb. MATH

The following will be useful for the next problem: If $f $ is a differentiable function of $\theta $, then the slope of the tangent line to the graph of $r=f(\theta )$ at the point $(r,\theta )$ is MATH provided that MATH at $(r,\theta )$.

Problems

3. Graph the following polar equation and find all points of horizontal tangency (that is, find the values of $\theta $ such that MATH and MATH).

MATH

4. Graph the following polar equation. Find an interval for $\theta $ over which the graph is traced only once.

MATH

5. Graph the following polar equation so that you get the curve depicted below. The curve is given by MATH. Over what interval must $\theta $ vary to produce the curve?


3b10__61.png

6. Heart to Bell Graph the polar equation MATH for $0\leq \theta <\pi $ for the integers $n=-5$ to $n=5$. What values of $n$ produce the "heart" portion of the curve? What values of $n$ produce the "bell"?

The following will be useful for the next problem: If $f$ is continuous and nonnegative on the interval $[\alpha ,\beta ]$, then the area of the region bounded by the graph of $r=f(\theta )$ between the radial lines $\theta =\alpha $ and $\theta =\beta $ is given by MATH MATH $r^{2}d\theta $.

Problems

7. Graph the following polar equation and find the area of the indicated region.

a. Inner loop of $r=1+2\cos \theta $.

b. Between the loops of $r=1+2\cos \theta $

c. Inside $r=3\sin \theta $ and outside $r=2-\sin \theta $

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