# 8.3 Polar coordinates

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In this section, you will:
• Plot points using polar coordinates.
• Convert from polar coordinates to rectangular coordinates.
• Convert from rectangular coordinates to polar coordinates.
• Transform equations between polar and rectangular forms.
• Identify and graph polar equations by converting to rectangular equations.

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.

## Plotting points using polar coordinates

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] .

## Plotting a point on the polar grid

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 .

## Plotting a point in the polar coordinate system with a negative component

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, $\text{\hspace{0.17em}}r=-2.\text{\hspace{0.17em}}$ We can approach plotting a point with a negative $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ in two ways:

1. Plot the point $\text{\hspace{0.17em}}\left(2,\frac{\pi }{6}\right)\text{\hspace{0.17em}}$ by moving $\text{\hspace{0.17em}}\frac{\pi }{6}\text{\hspace{0.17em}}$ in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
2. Move $\text{\hspace{0.17em}}\frac{\pi }{6}\text{\hspace{0.17em}}$ in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

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.

## Converting from polar coordinates to rectangular coordinates

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 $\text{\hspace{0.17em}}\theta .$

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena