For more information, please see full course syllabus of Math Analysis
For more information, please see full course syllabus of Math Analysis
Polar Equations & Functions
- We can set up equations and functions using r and θ exactly the same way as we did for x and y. Generally, θ is the independent variable (like x was), while r is the dependent variable (like y was).
- We always assume that θ is in radians ([(π)/2], [(7π)/4], etc.).
- We graph polar equations/functions in the same way we graph "normal" stuff: plug in values, plot points, and connect with curves to make a graph. Since θ is the independent variable, we plug in some value for it, then see what distance r we get out at that angle.
- Just like graphing with rectangular equations, you don't need to plot a huge number of points-merely enough to sketch the graph. Many polar equations involve trigonometric functions. The "interesting" points are when the trig function produces a zero or an extreme value (sin, cos ⇒ ±1). Figure out where these interesting values will occur and use them to help plot the graph.
- If at anytime you're unsure how the graph will behave, just plot more points. The easiest way to work out your uncertainty is by calculating more points.
- Occasionally you will see an equation that only uses one variable: that's fine! It means that variable is fixed, while the other can change freely.
- Sometimes we'll want to convert an entire equation or function from polar to rectangular, or vice-versa. We can do so with the same conversion formulas we figured out and used in the previous lesson. Since these formulas were based off any x, y, r, θ,
they work fine for substituting in equations.
- Polar ⇒ Rectangular: x = r cosθ y = rsinθ
- Rectangular ⇒ Polar: r^{2} = x^{2} + y^{2} tanθ = y/x
- If you have access to a graphing calculator, it's great to try graphing some polar equations with it. It's a new way of looking at graphing, so it helps just to play around. For more information, check out the appendix on graphing calculators.
Polar Equations & Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- Equations and Functions
- Graphing in Polar Coordinates
- Graphing in Polar - Example, Cont.
- Tips for Graphing
- Graphing Equations of One Variable
- Converting Coordinate Types
- Interesting Graphs
- Graphing Calculators, Yay!
- Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
- Check Out the Appendix
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:04
- Equations and Functions 1:16
- Independent Variable
- Dependent Variable
- Examples
- Always Assume That θ Is In Radians
- Graphing in Polar Coordinates 3:29
- Graph is the Same Way We Graph 'Normal' Stuff
- Example
- Graphing in Polar - Example, Cont. 6:45
- Tips for Graphing 9:23
- Notice Patterns
- Repetition
- Graphing Equations of One Variable 14:39
- Converting Coordinate Types 16:16
- Use the Same Conversion Formulas From the Previous Lesson
- Interesting Graphs 17:48
- Example 1
- Example 2
- Graphing Calculators, Yay! 19:07
- Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
- Check Out the Appendix
- Example 1 21:36
- Example 2 28:13
- Example 3 34:24
- Example 4 35:52
Math Analysis Online
Transcription: Polar Equations & Functions
Hi--welcome back to Educator.com.0000
Today, we are going to talk about polar equations and functions.0002
In the previous lesson, we introduced a new way to look at location in the plane, polar coordinates.0005
By now, we have seen thousands of graphs (probably literally), where x and y are in relationship with each other.0010
As the horizontal changes, the vertical changes somehow.0018
Either x and y are both in an equation, and we graph the equation, and we figure out all of the solutions to that equation;0020
we graph it; or y is a function of x, and we graph the function--as x changes, how will y respond?0025
We can do the exact same thing with polar coordinates: the variables r and θ can be put in some sort of relationship,0031
and then we can graph the resulting polar coordinates.0037
Before you watch the lesson, make sure you watch the previous lesson (before watching this one).0039
You really, really need to understand polar coordinates on their own.0045
You have to be able to understand how polar coordinates make locations on the plane before any of this stuff is going to really make sense.0048
So, if you are having difficulty with working through polar equations, but you don't have a really good understanding0056
of polar coordinates yet, that is the thing--you really want to work on polar coordinates.0060
Make sure you have watched the previous lesson before watching this one,0063
and that you are comfortable with polar coordinates before you try to work on polar equations and functions.0065
And if you can make sense of how polar coordinates work, polar equations and functions probably won't actually be that much harder.0070
All right, let's get started.0074
We can set up equations and functions using r and θ, exactly the same way as we did for x and y.0077
Generally, θ is going to be the independent variable, like x was.0082
Our x was allowed to change and vary around, and then r will be our dependent variable, in the same way that y was.0086
So, x was allowed to change around, and y responded to x's changes.0094
Similarly, here it is going to be θ that will be allowed to change around, and our distance r will respond to the angle that we are at.0098
Here are some examples; we could have an equation r = 1 + 2cos(θ) (we plug in a θ, and it tells us an r),0106
or a function r(θ) = 3sin(2θ), which is the same thing as...we plug in a θ, and it gives us what the r-value will be.0113
So, while r could be the independent, and θ the dependent, such relationships are pretty uncommon in polar equations and functions.0121
We normally think in terms of how length changes based on the angle--0132
if we go to this angle, what length it will be at--and not the opposite way,0136
just like when we are doing a rectangular graph, we normally think in terms of "for this horizontal location,0140
what height will I be at? For this horizontal location, what height will I be at?" and not0145
"For this height, what horizontal location will I be at?"0150
We don't normally think of height and then horizontal; we think of horizontal, and then height,0152
because x is the independent, and y is the dependent, just like here θ is the independent and r is the dependent.0156
We always, always assume that θ is in radians.0164
Whenever we are looking at our θ, it is assumed that θ is going to be in radians.0169
That is numbers like π/2, 7π/4, decimal things, etc.0173
It could be explicitly put in degrees, but that would be extremely rare.0177
Always, always, always use radians, unless you are being told explicitly otherwise, that this thing is in degrees.0182
It is extremely uncommon: you almost never see something like that.0189
I can't even think of one time I have seen it; so just don't expect that to happen--expect to be using radians when you are working with polar equations and graphs.0193
So, always think in terms of radians: you are plugging in radian values, and you are plotting with radian values on your angles.0202
All right, how do we graph in polar coordinates?0210
We graph polar equations and functions pretty much the same way that we graphed normal stuff.0212
You plug in some value for your independent (in this case, θ).0217
You plot the point that gets put out; and then you connect the whole thing with curves to make it into a graph.0220
So, θ is the independent variable; we plug in some value for it, and then we see what distance r would get out.0226
Over here, let's plot some points: if we are looking at the equation r = 1 + 2cos(θ), then r is going to come out once we plug in some θ.0231
So, if we plug in 0, well, 1 + 2cos(0)...cos(0) is 1, so 2 times 1 is 2, and 1 + 2 is 3.0239
So, we now have the point...not (0,3), but (3,0); that is one little confusing thing--0248
the fact that it is not (x,y); it is now (r,θ); so our independent variable is actually the thing that comes second.0255
So, don't let that confuse you.0261
Our distance out is 3, and our θ is going to be an angle of 0.0263
So, we end up getting this point right here; we have an angle of 0; we are at 0 above the starting location.0269
We haven't moved at all, and we are out on the third circle out; so we are at (3,0).0276
Next, π/4: if we plug in π/4, 1 + 2cos(π/4)...cosine of π/4 is √2/2; 2(√2/2) would be √2;0282
so that means that we get 1 + √2 out of this.0293
1 + √2...we figure that out with a calculator, so we can actually plot something down.0296
That is approximately 2.41; so at this point, we are at angle π/4, so we are on this arc sector line right here.0300
Notice, it is broken into eight pieces, so each one of them is going to be π/4, because up here is π/2.0309
So, we are at the line of π/4 angle; and then we go out 2.41: (2.41,π/4) is the point that we get out of this.0315
So, we are at 2.41 out, somewhere that looks around 2.4, a little more than 2, but even more less than 3.0328
So, we get this point right here; OK.0336
The same thing if we plug in π/2: if we plug in π/2, 1 + 2cos(π/2)...cosine of π/2 is 0, so we just get 1.0339
So, we have the point (1,π/2); and that gets us this point here.0348
So now, we think about how these things are going to connect through curves.0354
If we are really confused about it, we could just plot down more points.0357
We could put down π/6 and π/3, as well.0360
And if we want even more points, we could continue to plot down more and more points.0362
But we would probably be able to get a pretty good sense with just π/6 and π/3.0366
But we can even just figure this out by thinking, "Well, how is it curving--what is happening?"0370
So, as the angle goes up, notice that, because it is cosine of θ, as this goes to larger and larger values,0373
0 to π/4 to π/2, cosine shrinks down and down.0381
So, this portion of our equation, the 2cos(θ), will get smaller and smaller as cos(θ) gets smaller and smaller.0384
So, as we get larger and larger, it shrinks down to ultimately this value of 1 right here.0391
We go up, and as it gets farther out, it shrinks down; it shrinks down; it cuts through here; it shrinks down; it shrinks down.0396
And we get something like that curve.0403
We can see it precisely here, now drawn by a computer: r = 1 + 2cos(θ).0405
At this point, we just keep plotting more points.0411
Here is a new, interesting one to think about: if we plotted 3π/4, well, that is 1 + 2cos(3π/4).0413
What does 3π/4 come out to be? That is 1 + 2(-√2/2), so that gets us 1 - √2.0421
We approximate that with a calculator; we get -0.41, which means that we have the point (-0.41,3π/4).0430
So, here is our 3π/4; it is right here; this is the 3π/4 location.0440
We are on this line; but we have -0.41, so we are going in the opposite direction, and we are here for our point.0447
We plug in π; 2 times cosine of π; what is cos(π)? It is -1.0455
So, 2 times -1 get us -2; 1 - 2 gets us -1; so we have gotten the point (-1,π) out of this.0462
So, at the angle of π (here is angle π), we are going to be going opposite the length of 1, and so we are here, as well.0470
So, once again, it is continuing to drop down; this part here is continuing to get smaller and smaller, until eventually it becomes negative.0478
It actually continues out to this curve here, and then it cuts through.0485
It will always have to cut through the origin, because if it gets a 0 in the r, then it has to go through the origin,0488
because distant 0 from the origin, distant 0 from the pole, means at the pole; cut through here.0493
And then, curve up to here; and that is what we have so far.0500
It is computer-drawn there, so it is a little more accurate; but we basically just keep plotting points.0503
If we plug in 5π/4, we see that that is at -0.41; so at 5π/4, that would be this angle here.0508
So, we are in the opposite now; we are over here at 3π/2; we have length 1, so at 3π/2, we are here; length is 1 in that direction.0515
7π/4: we are at a length of approximately 2.41, so at 7π/4...we are at 2.41 from here.0523
And we continue this curving; we can also see at this point, perhaps, that is going to end up being symmetric.0529
If we looked at the entire table of values, we would see that there is some symmetry going on in the way these distances are coming out,0534
and that we are going to end up seeing the top part...this curve so far will happen and just sort of flip over;0539
and we will end up getting it out like this.0547
And finally, here is a computer-drawn version that is better than my slightly-imperfect drawing.0553
And that is what we end up getting out of this.0559
You plot points; you work it out; you graph the whole thing.0560
Just like graphing with rectangular equations, you don't need to plot a huge number of points.0564
It won't hurt; if you plot more points, it is not going to hurt.0568
But you really only have to plot enough so that you can sketch the graph.0571
Which points should you plot--what are the useful points to plot--what are the interesting points to plot?0575
Many polar equations involve trigonometric functions; the interesting points are when the trigonometric function produces a zero,0580
when we plug some θ into that trigonometric function and it puts out 0,0588
or when it puts out an extreme value (for sine and cosine, that is positive or negative 1).0592
Plugging in those values for our θ lets us see what the most extreme values are that our function is going to make,0597
which helps us have an understanding of how the whole thing is behaving.0602
More points will just make it smoother and easier to make the curves.0605
But those are the most important ones of all; so you definitely want to make sure that you are plotting those.0608
If we want to try to plot these extreme values and these 0's, we want to figure out where these interesting values will occur.0613
How are we going to get to it? Trigonometric functions tend to have a pattern.0618
We are used to working with sine and cosine; so we want to think in terms of sine--0621
we have to plug in 0 or π/2 or π; if it is sin(θ), we have to plug in 0 or π/2 or π or 3π/2...0625
any of these would end up getting an interesting number.0633
Sin(0) is 0; sin(π/2) is 1; sin(π) is 0; sin(3π/2) is -1; we are used to working with that from all of our work in trigonometry.0635
If we go to something else, though, like cos(5θ), well, it is not just θ anymore; it is 5 times that.0644
So, if it were cosine of something, we would want that something to be 0, and then π/2, and then π, and then 3π/2, and then 2π, and so on.0651
But in this case, it is 5 times θ; so that has to be taken into account in how our thing works.0658
So, the 0 is still just going to be the same; if we plug in a 0, 5 times 0 still gets a 0.0663
But if we wanted to try to figure out π/2 = 5θ, what does our θ have to go in to get that interesting first value of π/2?0669
We divide both sides by 5, and we would get π/10 = θ.0677
π/10 = θ, because if we plug in π/10, 5 times π/10 is π/2; cos(π/2) gets us 0.0682
So, that is our next interesting thing.0688
Similarly, it is going to continue on this pattern of π/10 being the interval here.0690
π/5 (I said that the wrong way...) 5 times π/5 would get us π; cos(π) is -1, an interesting value.0695
The cosine of 5 times 3π/10 would get us cos(3π/2), which is 0, an interesting value.0704
So, we are thinking in terms of what we have to plug in here, total, to get 0, π/2, π, 3π/2, 2π, the normal, interesting values.0711
But then, we have to pay attention to the fact that it is not just θ; it is something interacting with θ.0720
So, we have to pay attention to what the numbers are going to be.0725
And that helps us figure out what numbers we want to plot in when we are trying to graph.0727
Finally, what if it was 2θ + 1? In this case, θ = 0 won't even show up, because we have to figure out what would make this something.0732
What would make 2θ + 1 turn into 0?0739
Well, if we plug in -1/2 for θ, 2 times -1/2 gets us -1; -1 + 1 gets us 0.0742
So, if we plug in -1/2 for our θ, we get out a 0 from it.0752
What if we plug in θ = π/4 - 1/2?0756
Well, then we have 2θ + 1; so 2(π/4) - 1/2 gets us π/2 - 1 + 1; so we get π/2 out of this.0759
Let's work that one out: if we had 2θ + 1 = π/2, we would have to work out how to get to this.0768
2θ = π/2 - 1...θ...if we are trying to get to this value of π/2,0777
if we want this whole something here to come out to be π/2, the θ will have to end up being π/4 - 1/2.0784
And that is where we are getting it.0792
It is the same thing if we want this whole thing to be π; we end up needing π/2 - 1/2.0793
We want the whole thing to be π/2; we need to plug in 3π/4 - 1/2 for our angle θ.0798
Notice that, in each one of these, we are stepping up by π/4 each time.0804
Over here, we were stepping up by π/5 each time.0808
Once you start to notice the pattern, the pattern will normally continue.0811
But it will depend on the specific circumstances of what is inside of your trigonometric function.0814
Beyond looking for these interesting points, you might notice repetition in trigonometric functions leading to symmetries in the graph.0820
Depending on the way the trigonometric functions are set up, you might notice that all of the same stuff is going to happen here.0825
And because of the way that my angle is going to work, there is going to end up being a symmetry occurring in the graph.0830
If you notice the symmetry, if you see that a symmetry will certainly occur, just use that to make graphing easier.0835
You won't have to plot all of those points, because you will see0839
that it is just going to end up doing effectively the same thing, but reversed or flipped in some way.0842
If at any time you are unsure how the graph will behave, just plot more points; that is the easiest way to be sure of what is going to happen of all.0846
If you are uncertain about what will come next, calculate a bunch of points.0854
Just drop them in and connect the curve through those points.0859
The more points you have down, the easier it will be to see how the curve works.0861
After time, you will develop a sensibility; you will get an intuition for how these curves are going to come out--how they are going to look.0865
You won't have to plot as many points.0871
But when you are just starting out, if you are uncertain about a graph, plot more points, and you will have a good idea of where to go.0873
All right, occasionally you will see an equation that only uses one variable, where there is just one variable there.0879
A lot of students get scared; there is no reason to get scared.0884
It is totally fine to have one variable; it just means that that variable that you said is fixed, and the other one can change freely.0887
For example, if we had r = 2, then that means our distance is always 2.0893
But θ--does θ show up in r = 2? θ doesn't show up at all.0898
So, θ can be anything over here; in our red one over here, θ can be anything.0902
So, if θ can be anything, then that means we set r = 2; so it is going to always be on this length of 2, this distance of 2 from the center.0908
But our θ can end up going to any spin at all--it can be anything.0916
So, as θ is allowed to spin positive or negative, it is going to always end up being stuck on the circle.0921
So, we end up drawing this perfect red circle at a distance of 2 from the origin.0926
A similar thing is going on over here with θ = π/3; if θ = π/3, we never mentioned r in there.0932
We never mentioned the distance, so that means r can be anything.0937
Since r can be anything, we have to be on the angle of π/3.0941
But r could be any positive thing, so it could go out forever this way.0947
And r could be any negative thing, so it could go out in the opposite direction.0950
So, with θ = π/3, with θ at a fixed value, we end up creating a line.0953
With r at a fixed value, like r = 2, we end up creating a circle, because either with r (we fix r) we have fixed a distance,0959
but we are able to go to any angle (so we are making a circle); or if we fix an angle,0966
and we are allowed to go to any distance, we have drawn a line.0970
And that is what is going on if we just fix a single variable.0972
Converting coordinate types: sometimes, we will want to convert an entire equation or function from polar to rectangular, or vice versa.0976
We can do so with the same conversion formulas we figured out and used in the previous lesson.0983
So, when we figured out these formulas, they were based off any x, y, r, and θ.0987
We didn't say what x, y, r, and θ had to be; so this is going to work for converting the variables in equations.0992
They can be used to convert from one variable type to the other variable, to convert from (x,y) to (r,θ) or (r,θ) to (x,y).0998
So, previously we had x = rcos(θ), y = rsin(θ) for one set.1004
And then, the other set of formulas was r^{2} = x^{2} + y^{2}; tan(θ) = y/x.1009
Pretty much any way that you can see these working out to allow you to swap variables around,1014
so you can get to the kind of variables you want--go ahead and use it.1018
These are the formulas that will allow you to convert from one type of equation to another type of equation,1021
to switch from polar to rectangular or rectangular to polar.1026
And if you ever forget any of these formulas, if you forget, you can re-derive them by drawing this picture right here,1029
because we know that r is always the hypotenuse, and we know that x and y are the horizontal and vertical;1036
and we know that θ has to be the angle inside of the triangle.1043
So, we can figure out all of these equations: x = rcos(θ); y = rsin(θ);1046
r^{2} = x^{2} + y^{2}, tan(θ) = y/x,1051
just through basic trigonometry, because we know that (right down here) it is a right triangle.1054
So, you can just use basic trigonometry and re-derive this if, in the middle of an important exam, they all disappear from your head.1058
You can just draw a picture and do basic trigonometry; it is not too hard.1063
All right, polar equations allow us to make really interesting graphs--that is to say, crazy, bizarre, cool, strange graphs--1068
graphs that look nothing like the kind of graphs that we are used to with rectangular equations.1074
They allow us to make really, really interesting stuff very easily.1080
For example, look at this red one right here: could you imagine figuring out any way to graph that with rectangular x and y,1083
to make an x and y equation that could make this flower-looking thing?1089
That looks so unlike what we are used to graphing, but it only takes 2sin(6θ) + 0.5.1094
We are able to say that incredible thing, where it has these large petals and these small petals,1101
and a bunch of them repeating in this kind-of symmetrical pattern--with very, very little work.1105
Very, very little writing is required to be able to make this incredibly detailed picture.1110
The same thing over here in the blue picture: we are able to get this weird, squished thing that doesn't really look like anything specific.1114
But it is a picture, and it is not very hard to write out.1120
Once again, 2cos(θ) - sin(5θ)--we are able to get this very strange-looking picture,1124
that there would be no easy way for us to create a graph with x and y coordinates that we would be used to using here.1130
But in polar graphs, it is pretty easy to do.1136
We can make really interesting things, things that we were really not used to seeing before, with not that long of an equation.1139
Polar equations and functions are really a new way of thinking about graphing.1148
As such, it is a great time to use a graphing calculator.1151
This is the best time for graphing calculators: plot random equations in there.1155
Alter equations that you already understand, and just get a sense for how polar stuff works by playing around with graphs on a graphing calculator.1159
If you want more information, check out the appendix.1166
There is an appendix to the course entirely on graphing calculators--how to use graphing calculators,1168
what good graphing calculators are, and even if you don't own one, and you are not going to buy one--1172
absolutely, for sure--there are free options out there.1176
So, there is lots of cool stuff where you can go on the Web really quickly.1179
And in five minutes, you can be graphing polar stuff; probably in one minute, you can be graphing polar stuff.1183
And you would be able to get that without having to spend any money on a graphing calculator.1188
And just playing around with this stuff is going to help you understand polar graphs massively.1192
Trying different things, playing around, putting in things, looking at how changing one number here changed the whole thing--1199
just being able to see that incredible speed of responsiveness of a graphing calculator,1205
being able to change immediately when you do something...1210
You don't have to take all of the time to plot it carefully, because that goes so slowly; it is hard to realize what is going on.1212
But if you change one variable in a graphing calculator, and it creates a new graph,1217
you will be able to gain this really beautiful intuition of how polar graphs work--how a polar equation creates the graph associated with it.1220
I really, really, strongly recommend: if you have a graphing calculator, and even if you don't have a graphing calculator,1228
check out the appendix; I will talk about lots of places where you can get free ones,1233
where you can just go and play with them right now, immediately.1236
And you will be able to get a really good understanding of how polar graphs work with not that much effort.1239
Just playing around for 10 or 15 minutes will give you such a better understanding of polar graphs, if you find them difficult, even in the slightest.1244
Also, I want to just point out one really important thing.1250
When you are using a graphing calculator, pay attention to the interval that θ has.1253
Normally, they are going to start your θ being from 0 to 2π.1256
And that is going to often be enough to give you the entire graph; but it won't always.1261
It might not show you everything; so if it doesn't show you everything you need to see--1265
if there is some part missing (you might want to expand it, and you might not even realize that there is something missing)--1270
you might want to just try increasing your interval to -6π, +6π, or -10 to +10.1274
And just see if that changes the graph.1280
If it doesn't change the graph, then you know that a 0 to 2π interval is enough.1282
But if it does change the graph, that tells you that you need to think about how big your interval needs to be, to be able to see the whole graph.1285
And maybe it won't even be possible to graph the entire interval all at once on one graph, like we are going to see in Example 2.1290
All right, we are ready for some examples.1296
First, let's graph the function r(θ) = 3sin(2θ).1297
We plug in θ's here; it gives some value here, and that tells us what our r is going to be.1302
It is just plugging in a number and getting something out of it; and that tells us our r.1307
We plug in some value for θ; we get r.1312
Let's figure out some values for this; let's figure out how often we need to do this.1314
Well, we have 2θ as our thing in here: 2θ...if we were solving for the very first interesting point,1319
which would normally occur at π/2 (0 is an interesting point, but we can be certain that 0 is already going to show up,1324
and that one doesn't tell us as much), then that is going to be θ = π/4.1330
So, we are going to have interesting points occur at an interval of every π/4 that we go out.1334
Let's plug in π/4 to figure this out.1339
We will plug in θ's; we are going to make a whole bunch of them.1342
And here are our r's; and if we plug in 0, 3sin(0)...sin(0) is 0, so we are just going to get 0 in here.1350
If we plug in π/4 (we figured out that our first interesting point was π/4), 3sin(2π/4)...2π/4 is π/2; sin(π/2) is 1; 3 times 1 is 3.1362
So, we get +3 over here for our r.1372
At this point, we can draw a graph; so we can start plotting some things.1374
We know that we are going to have to at least get out to a distance of 3.1378
And I will tell you that it actually is only going to go out to a total distance of 3, at the maximum.1381
So, we will have circles every 3; so here is our first distance circle...1385
Pardon me if my circles aren't absolutely perfect; I am but a human.1392
The second distance circle; and our third distance circle...OK.1396
And then, let's see: since we know that we are going to be based off of π/4, let's cut angles at π/4, as well; cool.1416
So now, we can plot some points: at an angle of 0, we go out 0, so our first point is just at the pole, on what we used to call the origin.1428
At an angle of π/4, this one right here, we are a distance of 3: 1, 2, 3 distance out.1438
Next, let's try π/2, the next π/4 forwards, the next interesting point.1445
We plug in sin(2π/2); 2π/2 is π; sin(π) is 0, so we end up getting 3(0) is 0.1450
By the time we get back to π/2, this angle here, we are back down to 0; what does that look like?1458
As π/4 goes to π/2, it goes down; from 0 up to π/2, we increase to 3, and then we decrease back down to 0.1464
We increase to 3; and then we decrease down to 0; from 0 to π/2, we go up to 3, and then down to 0.1473
So, as we spin counterclockwise, our thing increases; we touch that, and we spin back down.1481
We get smaller and smaller, back down to 0; so that is the first part of our graph.1491
Let's see what else is going to happen here: if we plug in 3π/4, sin(2(3π/4))...2(3π/4) is 3π/2;1495
sin(3π/2) is -1; 3 times -1 is -3; so at an angle of 3π/4, which is here, we are at -3.1506
So, we are going to go in the opposite direction; we are going to go opposite.1515
We were going this way, but we are now going to go opposite, because it is -3.1518
So, we are out 1, 2, 3 here; and then, at π, the next interesting place, sin(2π) is 0;1521
2 times π is 2π, so sine of 2π is 0; so we get 0 once again here.1528
And so, it is going to do the same thing, where, as it goes from π/2 to π, it becomes...1532
here is 0; here is π/2; here is π; so for the first part, it went up to 3, and then it went back down to 0,1538
when it got to π/2; and now it is going to go down to -3, and then it is going to go up to 0.1544
So, we are seeing it go up, and then down, and then negative, and then back to 0.1550
So, for this part, it is going more and more negative; so we end up seeing it curve out like this as it gets to larger and larger angles.1554
We see it spin this way; OK.1561
Next, we have 5π/4; at 5π/4, we plug that in; 2 times 5π/4 is going to be 10π/4, which is the same thing as 2π +...1567
well, let's just write this out, because that way it is a little less confusing.1581
5π/4 times 2 is equal to 10π/4; and look, that is the same thing as 8π/4 + 2π/4.1583
So, that is the same thing as 2π + π/2; everything that we have here is just going to end up being the same thing as this.1599
The π/4 here will match up to the 5π/4 here; so we are going to end up getting 3.1608
With this idea in mind, we could actually realize, really quickly, that π/2 here is going to match to 6π/4, which is 3π/2.1613
So, as we go that extra π forward, because we have this 2 here, it is going to end up repeating everything, as well.1622
Our angles are going to be new and different and interesting, but the r's will end up repeating.1627
We are going to see a repeat of this part here; it is just going to repeat here.1631
So, 3π/2 gets us 0; 7π/4 will get us a -3; and at 2π, we will be right back where we started at 0.1635
So, it will end up getting back to where we started.1644
Let's plot 5π/4; at an angle of 5π/4, we are at this part; we go out a distance of 3, so we are here.1647
So, we just got back to the origin when we were at π; the same thing--it curves out and grows larger when it gets out to it.1653
And then, it gets smaller and smaller at this point; it drops back down.1659
We are starting to see some symmetry; it is like there are petals here.1662
"Petals" is a way of talking about this.1665
At 7π/4, we are here; but once again, it is -3, so we go in the opposite direction.1667
We go out to here, and there we go--they should be perfectly symmetrical, if the graph was absolutely perfect.1675
If it was drawn by a computer, they would end up being perfectly symmetrical petals.1684
But we can get a pretty good sense of what is going on, even drawing it by hand.1687
The second example: Graph the equation r = 1.5^{θ/2}, where θ is between -2π and 2π; it goes from -2π to 2π.1692
At this one, let's start...we will graph...well, we really have no idea of what this is going to end up being.1702
We have 1.5r = 1.5^{θ/2}; we are dividing our θ by 2.1708
So, we know that we are going to want 0 in there, because it is right in between -2π and +2π.1719
And we want to go all the way down to -2π and all the way up to 2π.1724
So, let's do this by π/2; π/2 will be easy to graph, as well, since they are the cross-axis that we normally have in there.1727
A distance of r...if we plug in θ at 0 first, well, 1.5 raised to the 0--that is easy.1734
We always end up getting 1; if you raise any number to the 0, you end up getting 1 out of it.1741
Next, though, this is a little bit more difficult: if we plug in π/2, then what do we get out of this?1746
Well, we will have to use a calculator; so here is how we do the first one.1752
If we were to try to figure out 1.5, raised to the π/2, over 2...we have π/2, over 2...1754
well, that is going to be the same thing as 1.5 to the π/4.1762
We might not be able to figure this out by hand, but we could put this into a calculator: 3.14...1767
1.57 would be half of that; so it is just going to be plugging in those numbers and getting an approximate value.1772
So, this comes out to be approximately...using a calculator, we get 1.37 out of it.1779
Sorry, I meant to say 1.57, because 3.14/2 would be equal to 1.57.1784
But then, this is 3.14/4, so that is 1.57/2; the point is that we could work it out with a calculator1790
and get an approximate decimal value for what that ends up being.1799
That comes out to be 1.37; we do the same thing for each one of these.1802
We plug in π; so it is 1.5 to the π/2 or 1.5 to the approximately 1.57.1806
We raise that...our calculator gives us that that is approximately 1.89.1813
3π/2...our calculator gives us that that gives us approximately 2.60...2π...we get approximately 3.57.1818
What if we went the other way? -π/2...raising that up there...1.5^{-π/2} divided by 2, so 1.5^{-π/4}...1829
we get approximately 0.72; raising it to the -π/2...we get 0.53; raising it to the -3π/2, over 2, gets us 0.38.1837
And finally, at -2π, we get 0.28.1852
So notice; at 0, if we take 0, and we go up with 0, if we go up, the value gets larger and larger.1857
We have it getting larger and larger faster and faster, because remember: this is an exponential function.1864
So, it will get faster and faster growing as we go to larger and larger values of θ.1869
As we go to negative values of θ, though, it gets smaller and smaller, because it is an exponential function.1874
Once again, we are seeing the tail part get really down close to that x-axis.1877
So, we can draw this out: the most extreme value that we get out to is 3.57.1882
So, I will set our most extreme circle at a distance of 4.1889
These single cross bars will be enough, because we are only concerned1893
if π/2 is the only real reference angle we have going on here; so that will be OK.1896
So, here is a distance of 1 circle; here is a distance of 2 circle; here is a distance of 3 circle; here is a distance of 4 circle.1901
Oops, that got a little bit out of hand; here is a distance of 4 circle; that is better.1916
OK, that is a pretty reasonable setup for our axes.1929
At this point, we can plot some points: at 0, at an angle of 0, we are 1 out; so here is our first point.1932
At an angle of π/2, going straight up, we are at 1.37; so we are almost to halfway out.1940
At π, this angle here, we are at 1.89, getting pretty close to that distance of 2.1949
At 3π/2, we are at 2.60, a little bit over halfway between the 2 and the 3 ring.1956
At 2π, we are at 3.57, a little bit over halfway between the 3 and the 4 ring.1962
We have this...as we go to a larger and larger angle, the distance out increases.1968
We see it spiraling out, the farther out it gets.1973
It continues to spiral out this way; so this is what we see as the angle gets larger and larger--it spirals out.1977
What if the angle goes negative? Well, at -π/2, remember, this is -π/2,1986
because it is talking about going clockwise instead; so at -π/2, we have 0.72.1991
0.72 is around here; at -π, that is here, because remember: it is clockwise now;1997
we are at 0.53; at -3π/2 (that is this one here), we are at 0.38; at -2π, we are at 0.28.2005
So, we end up seeing it continue to spiral in and in and in and in and in.2018
-2π to 2π: that is what is stopping this from continuing out forever.2024
If this was allowed to keep going forever, we would see it spiral off way out forever.2027
If this was allowed to continue forever, we would see it spiral into the center more and more and more and more.2032
So, that -2π to 2π--that is why we have to have an interval set--because sometimes,2036
if we don't set an interval, we could just keep going forever.2041
All right, and that is why we have it set in our graphing calculator, if you are using a graphing calculator.2044
You have to pay attention to the interval, because sometimes it will end up cutting off parts of the graph that you want to see.2047
You wouldn't see that part where it gets to continue to spin in if you didn't have a larger than 0 as the bottom of your interval.2052
You have to go to -2π, -4π, -10π...to really get a sense of just how much that spirals into the center.2057
All right, the third example: convert the equation from polar to rectangular, then solve for y in terms of x.2064
What were all of our equations? We have x = rcos(θ) as one of our formulas for changing.2070
y = rsin(θ) is another formula; r^{2} = x^{2} + y^{2} is another; and tan(θ) = y/x.2076
So in this case, we see right away that 6r^{2} times cos^{2}(θ)...2086
we are not quite sure about that part, but here is our sin(θ), and here is our sin(θ).2091
So, we can swap them out; that is 3y = 7.2096
What about this part here--what about r squared, cosine squared, θ?2099
Well, we realize that that sounds an awful lot like r cosine θ; so how can we get rcos(θ) to show up there?2103
We maybe think, "Oh, well, there is r^{2}; there is cosine squared; we can pull that squared out, and we could write this as 6rcos(θ)."2109
And then, that whole thing is squared; and then, minus 3y = 7.2117
At this point, we can swap out x = rcos(θ); we have 6 times x^{2} - 3y = 7.2122
Add 3y; subtract 7 on both sides; so we have 6x^{2} - 7 = 3y.2131
We can divide by 3 on both sides; we could write this as y = 6x^{2} - 7, and that divided by 3.2137
We have managed to convert this to a rectangular x and y, and it is now in the form y = stuff involving x.2145
What if we are doing the reverse? We are going from rectangular to polar.2153
Last time, we went from polar to rectangular; now we are going the other way.2157
Solve for r in terms of θ; once again, we have the same x = rcos(θ), y = rsin(θ);2160
r^{2} = x^{2} + y^{2}; and tan(θ) = y/x.2169
In this case, we see that that is nice; we have rcos(θ), rsin(θ)...we have y; we have x.2176
So, we can just swap those out directly; we can swap them out for what we have here.2182
We swap them out; y is rsin(θ)...equals 2 times...x is rcos(θ), so 2 times rcos(θ), plus 3.2187
At this point, we were told to solve for r in terms of θ, so we need to get our r's on one side, so we can get just r by itself.2199
We move the 2rcos(θ) over by subtracting it, so we have rsin(θ), now minus 2rcos(θ), from both sides;2207
that equals...+3 is still left over here; now, notice: we have an r here and an r here, so we can use the distributive property in reverse;2214
we pull that r out, and we have sin(θ) - 2cos(θ) = 3.2222
We now divide that out; so we have just r; so r = 3/(sin(θ) - 2cos(θ)).2232
That might be a little surprising; that seems like a fairly complicated thing, if it is just going to give us a line.2243
But some things...polar is better at graphing certain kinds of pictures, and rectangular is graphing other kinds of pictures.2248
So, it depends on things; rectangular is great for graphing lines; polar is not as great at graphing lines.2255
And you might be surprised that that would even end up coming out to be a line.2259
Try plugging it into a graphing calculator, and you will see that that ends up giving us y = 2x + 3.2263
It is just another way of graphing it.2267
You might have a little bit of difficulty seeing why that ends up giving it.2269
If you think about it, this bottom part here is going to sort of work as an asymptote,2272
as it approaches the same angle that this line is based on, which is why it is going to shoot off infinitely in both the top and the bottom.2276
So, think about that for a while; try graphing it.2283
Just in general, try to play around with graphing as many polar functions as you can.2285
It will really give you such a great sense of how the stuff is working if you just play with it for a while on a graphing calculator.2289
All right, we will see you at Educator.com later--goodbye!2294
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