Objectives
- To understand the measurement of and conversions from polar coordinates
- To practice graphing polar inequalities
Section 1.5 Polar Points and Regions
Subsection 1.5.1 Polar Coordinates
Recall that polar coordinates gives us an alternative way of representing points, curves, and regions in two dimensional space. In place of \((x,y)\) we use \((r,\theta)\text{.}\) Here \(r\) represents the distance from the origin and \(\theta\) is the angle made moving counterclockwise from the positive \(x\)-axis.
Activity 1.5.1.
(a)
Consider the points given in the following figure. For each labeled point give the polar coordinates of the points in the form \((r,\theta)\text{.}\)
(b)
Use the equations \(x = r \cos(\theta)\) and \(y = r \cos(\theta)\) to convert the above points to \((x,y)\) coordinates.
Activity 1.5.2.
(a)
Consider the polar regions given by the various colored areas in the following figure:
For each region fill in values in the blanks
\begin{equation*}
\text{___} \leq r \leq \text{___} \qquad \text{___} \leq \theta \leq \text{___}
\end{equation*}
that represent each shaded region.
(b)
Shade in the polar region given by the following
- \(\displaystyle r \leq 2, \qquad 0 \leq \theta \leq \frac{\pi}{3}\)
- \(\displaystyle 1 \leq r \leq 2, \qquad \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}\)
- \(\displaystyle r \leq 1, \qquad \frac{5\pi}{4} \leq \theta \leq \frac{7\pi}{4}\)
Use the tool below to confirm your results.
