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Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see [link] ). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.
When we think about plotting points in the plane, we usually think of rectangular coordinates $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates , which are points labeled $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole , or the origin of the coordinate plane.
The polar grid is scaled as the unit circle with the positive x- axis now viewed as the polar axis and the origin as the pole. The first coordinate $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is the radius or length of the directed line segment from the pole. The angle $\text{\hspace{0.17em}}\theta ,$ measured in radians, indicates the direction of $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ We move counterclockwise from the polar axis by an angle of $\text{\hspace{0.17em}}\theta ,$ and measure a directed line segment the length of $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ in the direction of $\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Even though we measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ first and then $\text{\hspace{0.17em}}r,$ the polar point is written with the r -coordinate first. For example, to plot the point $\text{\hspace{0.17em}}\left(2,\frac{\pi}{4}\right),$ we would move $\text{\hspace{0.17em}}\frac{\pi}{4}\text{\hspace{0.17em}}$ units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in [link] .
Plot the point $\text{\hspace{0.17em}}\left(3,\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ on the polar grid.
The angle $\text{\hspace{0.17em}}\frac{\pi}{2}\text{\hspace{0.17em}}$ is found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in the $\text{\hspace{0.17em}}\frac{\pi}{2}\text{\hspace{0.17em}}$ direction, as shown in [link] .
Plot the point $\text{\hspace{0.17em}}\left(2,\text{\hspace{0.17em}}\frac{\pi}{3}\right)\text{\hspace{0.17em}}$ in the polar grid .
Plot the point $\text{\hspace{0.17em}}\left(-2,\text{\hspace{0.17em}}\frac{\pi}{6}\right)\text{\hspace{0.17em}}$ on the polar grid.
We know that $\text{\hspace{0.17em}}\frac{\pi}{6}\text{\hspace{0.17em}}$ is located in the first quadrant. However, $\text{\hspace{0.17em}}r=\mathrm{-2.}\text{\hspace{0.17em}}$ We can approach plotting a point with a negative $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ in two ways:
See [link] (a). Compare this to the graph of the polar coordinate $\text{\hspace{0.17em}}\left(2,\frac{\pi}{6}\right)\text{\hspace{0.17em}}$ shown in [link] (b).
Plot the points $\text{\hspace{0.17em}}\left(3,-\frac{\pi}{6}\right)$ and $\text{\hspace{0.17em}}\left(2,\frac{9\pi}{4}\right)\text{\hspace{0.17em}}$ on the same polar grid.
When given a set of polar coordinates , we may need to convert them to rectangular coordinates . To do so, we can recall the relationships that exist among the variables $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta .$
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