Plot points utilizing polar coordinates. Convert from polar works with to rectangle-shaped coordinates. Transform from rectangular coordinates to polar coordinates. Change equations between polar and also rectangular forms. Identify and also graph polar equations by converting to rectangle-shaped equations.

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Over (12) kilometers from port, a sailboat encounters turbulent weather and also is blown turn off course by a (16)-knot wind (see figure (PageIndex1)). How deserve to the sailor suggest his location to the shore Guard? In this section, we will investigate a an approach of representing ar that is different from a standard coordinate grid.

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Figure (PageIndex2)


Example (PageIndex1): plotting a allude on the Polar Grid

Plot the suggest (left(3,dfracpi2 ight)) top top the polar grid.

Solution

The angle (dfracpi2) is uncovered by sweeping in a counterclockwise direction (90°) from the polar axis. The suggest is located at a length of (3) systems from the pole in the (dfracpi2) direction, as shown in figure (PageIndex3).

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Example (PageIndex2): plot a point in the Polar Coordinate mechanism with a an unfavorable Component

Plot the allude (left(−2, dfracpi6 ight)) top top the polar grid.

Solution

We understand that (dfracpi6) is situated in the first quadrant. However, (r=−2). We can technique plotting a suggest with a an unfavorable (r) in two ways:

Plot the allude (left(2,dfracpi6 ight)) by moving (dfracpi6) in the counterclockwise direction and also extending a directed line segment (2) units right into the very first quadrant. Then retrace the directed heat segment ago through the pole, and continue (2) units right into the 3rd quadrant; relocate (dfracpi6) in the counterclockwise direction, and also draw the directed line segment from the pole (2) systems in the an unfavorable direction, into the 3rd quadrant.

See number (PageIndex5a). To compare this come the graph that the polar name: coordinates ((2,π6)) presented in number (PageIndex5b).

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Converting from Polar works with to rectangle-shaped Coordinates

When offered a set of polar coordinates, we might need to convert them to rectangle-shaped coordinates. To perform so, we can recall the relationships that exist amongst the variables (x), (y), (r), and ( heta).

(cos heta=dfracxr ightarrow x=r cos heta)

(sin heta=dfracyr ightarrow y=r sin heta)

Dropping a perpendicular from the allude in the plane to the x-axis forms a right triangle, as depicted in figure (PageIndex7). One easy way to psychic the equations above is come think that (cos heta) together the nearby side end the hypotenuse and (sin heta) as the the opposite side end the hypotenuse.

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Figure (PageIndex14): (a) Polar net (b) rectangular coordinate system

A similar discussion would demonstrate that the graph that the function (r=2 csc heta) will certainly be the horizontal line (y=2). In fact, (r=c csc heta) is the standard type for a horizontal heat in polar form, equivalent to the rectangular type (y=c).

See more: What Is An Autobiographical Narrative? ? What Are Autobiographical Narratives



Exercise (PageIndex5)

Rewrite the polar equation (r=2 sin heta) in Cartesian form.

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(x^2+y^2=2y) or, in the standard type for a circle, (x^2+(y−1)^2=1)