# 8.3 Polar coordinates  (Page 2/8)

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$\begin{array}{l}\begin{array}{l}\\ \mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{x}{r}\to x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \end{array}\hfill \\ \mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{y}{r}\to y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \end{array}$

Dropping a perpendicular from the point in the plane to the x- axis forms a right triangle, as illustrated in [link] . An easy way to remember the equations above is to think of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ as the adjacent side over the hypotenuse and $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ as the opposite side over the hypotenuse.

## Converting from polar coordinates to rectangular coordinates

To convert polar coordinates $\text{\hspace{0.17em}}\left(r,\text{\hspace{0.17em}}\theta \right)\text{\hspace{0.17em}}$ to rectangular coordinates $\text{\hspace{0.17em}}\left(x,\text{\hspace{0.17em}}y\right),$ let

$\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{x}{r}\to x=r\mathrm{cos}\text{\hspace{0.17em}}\theta$
$\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{y}{r}\to y=r\mathrm{sin}\text{\hspace{0.17em}}\theta$

Given polar coordinates, convert to rectangular coordinates.

1. Given the polar coordinate $\text{\hspace{0.17em}}\left(r,\theta \right),$ write $\text{\hspace{0.17em}}x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=r\mathrm{sin}\text{\hspace{0.17em}}\theta .$
2. Evaluate $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta .$
3. Multiply $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ to find the x- coordinate of the rectangular form.
4. Multiply $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ to find the y- coordinate of the rectangular form.

## Writing polar coordinates as rectangular coordinates

Write the polar coordinates $\text{\hspace{0.17em}}\left(3,\frac{\pi }{2}\right)\text{\hspace{0.17em}}$ as rectangular coordinates.

Use the equivalent relationships.

$\begin{array}{l}\begin{array}{l}\\ x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \end{array}\hfill \\ x=3\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{2}=0\hfill \\ y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ y=3\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{2}=3\hfill \end{array}$

The rectangular coordinates are $\text{\hspace{0.17em}}\left(0,3\right).\text{\hspace{0.17em}}$ See [link] .

## Writing polar coordinates as rectangular coordinates

Write the polar coordinates $\text{\hspace{0.17em}}\left(-2,0\right)\text{\hspace{0.17em}}$ as rectangular coordinates.

See [link] . Writing the polar coordinates as rectangular, we have

$\begin{array}{l}x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \\ x=-2\mathrm{cos}\left(0\right)=-2\hfill \\ \hfill \\ y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ y=-2\mathrm{sin}\left(0\right)=0\hfill \end{array}$

The rectangular coordinates are also $\text{\hspace{0.17em}}\left(-2,0\right).$

Write the polar coordinates $\text{\hspace{0.17em}}\left(-1,\frac{2\pi }{3}\right)\text{\hspace{0.17em}}$ as rectangular coordinates.

$\left(x,y\right)=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$

## Converting from rectangular coordinates to polar coordinates

To convert rectangular coordinates to polar coordinates    , we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

## Converting from rectangular coordinates to polar coordinates

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in [link] .

## Writing rectangular coordinates as polar coordinates

Convert the rectangular coordinates $\text{\hspace{0.17em}}\left(3,3\right)\text{\hspace{0.17em}}$ to polar coordinates.

We see that the original point $\text{\hspace{0.17em}}\left(3,3\right)\text{\hspace{0.17em}}$ is in the first quadrant. To find $\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ use the formula $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{y}{x}.\text{\hspace{0.17em}}$ This gives

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{3}{3}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{tan}}^{-1}\left(1\right)=\frac{\pi }{4}\hfill \end{array}$

To find $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ we substitute the values for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ into the formula $\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}}.\text{\hspace{0.17em}}$ We know that $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ must be positive, as $\text{\hspace{0.17em}}\frac{\pi }{4}\text{\hspace{0.17em}}$ is in the first quadrant. Thus

$\begin{array}{l}\begin{array}{l}\\ r=\sqrt{{3}^{2}+{3}^{2}}\end{array}\hfill \\ r=\sqrt{9+9}\hfill \\ r=\sqrt{18}=3\sqrt{2}\hfill \end{array}$

So, $\text{\hspace{0.17em}}r=3\sqrt{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{=}\frac{\pi }{4},\text{\hspace{0.17em}}$ giving us the polar point $\text{\hspace{0.17em}}\left(3\sqrt{2},\frac{\pi }{4}\right).\text{\hspace{0.17em}}$ See [link] .

## Transforming equations between polar and rectangular forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

Given an equation in polar form, graph it using a graphing calculator.

1. Change the MODE to POL , representing polar form.
2. Press the Y= button to bring up a screen allowing the input of six equations: $\text{\hspace{0.17em}}{r}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}.\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{6}.$
3. Enter the polar equation, set equal to $\text{\hspace{0.17em}}r.$
4. Press GRAPH .

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
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