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

find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what