Understanding Polar Coordinates and Conversion

In this video, we're going to go over a basic introduction into polar coordinates. We're going to talk about how to graph it, and a few other stuff as well. But let's go over the difference between rectangular coordinates... and polar coordinates. What do you think the difference is? Now you're familiar with rectangular coordinates. Basically it contains an x and y variable, whereas a polar coordinate contains r and theta, where r is the radius of a circle and theta is the angle measured from the positive x axis. So let's say if we wish to plot 3 comma 45 degrees. How can we do so? So r is 3 and the angle theta is 45. So let's make a graph. The first thing we should do is plot three circles. The first one is going to have a radius of 1, the second one is going to have a radius of 2, and the third one is going to have a radius of 3. So, we're going to draw the ray at a 45 degree angle, and it's going to go to the third circle, because it has a radius of 3. Now, keep in mind, a positive angle is measured from the x-axis rotating counterclockwise. So that angle is 45. And that's how you can plot a polar coordinate. Now, let's say if we wish to plot the point 2, 3 pi over 4. How can we do so? Now, if you're not sure where 3 pi over 4 is located, convert it to degrees. Let's multiply this by 180 divided by pi. The pi values will cancel. 180 divided by 4 is 45, and 45 times 3 is 135. So that point is equivalent to 2 comma 135. Now we know this is 0 degrees, this is 90, 180, and 270. So 135 is somewhere in quadrant 2. This is quadrant 1, quadrant 3, and quadrant 4. So we need to have an angle of 135, which is between 90 and 180, and we need to stop at the second circle, because the radius is 2. So this angle here, measured from the positive x-axis, that's 135 degrees, or simply 3 pi over 4. Now what about if r is negative? Let's say if we have negative 2 comma 60 degrees. How can we plot this particular point? So first, let's draw the circles. This is going to have a radius of 1, a radius of 2, and let's do one more with a radius of 3. Now we know that 60 is between 0 and 90, that's in the first quadrant. And let's plot 2, 60 first. And then we'll plot negative 260 so you can see the difference. So with a radius of 2, we need to stop at the second circle. And 60 is closer to 90 than it is to 0. So that's where this point is located. Now, if it's negative 2, you need to travel in the other direction. So, you need to travel over here. Notice that negative 2 is the same as positive 2 and 240. So, basically, you got to add 180 to this point. So, this angle here, that is 60. And this angle here is 60 plus 180, which is 240. So anytime r is negative, instead of going in the direction it should go, go 180 degrees in a direction opposite to where you should go. Let's try another example. Let's use... let's plot this point, negative 3, 120 degrees. So first, plot 3, 120. So that's going to be in quadrant 2 with a radius of 3. Now let's plot the other one, negative 3, 120. So we're going to travel directly in the opposite direction. So, it should be somewhere over there. So, if this angle is 120, 120 plus 80 is 300, so the other angle is 300, and this angle is negative 60. Sometimes, you may need to represent a point using other values. So, for example, negative 3 comma 120 is the same as... Positive 3, negative 60. If we had a negative angle, we would go in a clockwise direction as opposed to the counterclockwise direction. And it would lead us to the same point. Another way in which we can get the answer, we could say that it's also negative 3, 240. So instead of going 120 in this direction, we can travel negative 240 in this direction to get to this point. And because r is negative, it's going to flip back to this point. So to get 240, simply, this is negative 240 by the way, subtract 120 by 360, and that gives you negative 240. Now, we can get another positive r-value, which leads to the same point, if we add 360 to negative 60. So, it's going to be 3,300. So, instead of traveling negative 60 in this direction, we could travel 300 in this direction, also represented by the blue line. So sometimes you might be given an rtheta value, but you've got to find three other values that would lead to the same point. Two of them are negative, and two are positive. But typically, you might be given one out of those four options. So you'll be given one answer, and you've got to find the other three. Now let's say if we have the point 2, 30 degrees. And we wish to find the other three points that lead to the same terminal point. And the angle theta has restrictions. It's between negative 360 and positive 360. Find the other three polar coordinates. So the first one is r theta. The second one will still have a positive r value, but a negative angle. The third one will have a negative r value and a positive angle. And the fourth will have a negative r value and a negative angle. So we got the first one. It's 2, 30. To find the second one, r is going to be the same. But we're going to subtract 30 by 360. So that's going to be negative 330. So that gives us the second one. Now for the other two, r has to be negative. So we're going to make it negative 2. And what we need to do is we need to add 180 to our original angle. So 30 plus 180, that's 210. Now to find the other answer, we can either add 180 to negative 330, which is negative 150. Or, we can subtract 210 by 360, which will also give us negative 150. So, this one corresponds to that answer, where both r and theta are negative. And this corresponds to this one, where r is negative and theta is positive. Now, let's go ahead and plot the original point. So, here's the first circle, and here is the second one. So let's plot 230. So at an angle of 30, the ray has to stop at the second circle. So this is positive 30 degrees. Now, we can choose to travel in the other direction, and that would be negative 330, which is this answer. Now let's say if we want to use this point. Positive 210 ends right here. Now keep in mind, we're not going to use this ray because since r is negative, it's going to flip to this point, which is what we want. So that's negative, that's positive 210. Now for the other one, it's negative 150, which will lead us to the same point. And because r is negative, it flips back to the original terminal point. So that's how you can find the four polar coordinates that lead to the same terminal point. Try this example. Let's say r is 3. and the angle theta is 5 pi over 6. Take a minute and go ahead and find the other three values, the other three polar coordinates that lead to the same terminal point given that the angle theta is between negative 2 pi and 2 pi. So to find the first point what we need to do is we need to subtract 5 pi over 6 by 2 pi to get a negative angle with a positive r value. Now 2 pi is the same as 12 pi over 6, and 5 minus 12 is negative 7, so we have negative 7 pi over 6. So therefore, that's the second angle. Let's find the next polar point, but first, let's draw it. So this is the second circle, and this is the third circle. 5 pi over 6 is in the second... quadrant. It's over there. If you convert it to, let's say, an angle in theta, it's going to be 180 divided by 6, which is 30, because 18 divided by 6 is 3, add the 0, and then 30 times 5 is 150. So this is 5 pi over 6 traveling in that direction. And if we travel in this direction, we still have the same positive r value, but the angle is now negative 7 pi over 6. Now... Let's change 3 into negative 3. If we do that, we need to add pi to 5 pi over 6. So 5 pi over 6 plus pi is the same as 6 pi over 6, so that will give us 11 pi over 6. To get the other negative angle, let's subtract this one by 2 pi. So 2 pi is the same as 12 pi over 6. And 11 minus 12 is negative 1 pi over 6, or just negative pi over 6. So keep in mind, we need to get to this point, because when the angle is negative, I mean when r is negative, it's going to flip to the red line. 11 pi over 6 is in the fourth quadrant. So it takes us to this point, but because r is negative, it's going to switch to the red line. Negative pi over 6 will take us to the same initial point, and then the negative r will switch us back to the point where we want to be, which is here. So that's how you can find the four polar points given one of the points. Personally, I think this is helpful. If you know that it's going to be r theta, r negative theta, negative r theta, and negative r negative theta. So, let's say if you're given this point. To find the second point, simply, you're either adding or subtracting 2 pi. In this case, you're going to subtract it by 2 pi or 360. Since you want to get the negative angle. Now, from number 1 to number 3, you should either add or subtract by pi, or 180. So, if this angle is like 300, subtract it by 180. If it's like 60, add 180 to it. So, if it's more than 180, subtract it by 180. If it's less than 180, add 180 to it. Now, to find number 4, you can just find it from number 3. You can add or subtract by 2 pi. In this case, subtract by 2 pi since this is positive and you want the negative angle. Now, the next thing we need to go over is how to convert polar coordinates into rectangular coordinates. So you need to know that x is equal to r cosine theta and y is equal to r sine theta. So using those two equations, let's say if we have the point 4 comma 60 degrees. What is the value of x and y? So x is r cosine theta. So that's 4 cosine 60. And 60 degrees... is 1 half. So 4 times 1 half is 2. So the x coordinate is 2. Why is r sine theta? So that's 4 sine 60. Sine 60 is the square root of 3 over 2. So 4 divided by 2 is 2, and so it's 2 root 3. And that's how you can convert polar coordinates. into rectangular coordinates. Try this example 6,5pi over 6. Convert it into rectangular coordinates. So x which is r cosine theta that's 6 cosine 5 pi divided by 6 so what is cosine 5 pi over 6 keep in mind 5 pi over 6 is 150 anytime you see pi over 6 is 30 so 5 pi over 6 is 5 times 30 which is 150 cosine 150 is negative root 3 divided by 2 And 6 divided by 2 is 3, so this is negative 3 root 3. So now we've got to find the y-coordinate, which is 6 sine 5 pi over 6. Sine 150 is positive 1 half. And 6 times 1 half is positive 3. So the answer is going to be negative 3 root 3 comma 3. Now, what if we have rectangular coordinates? How can we find the value of r and theta? So, for example, let's say if we have the points 2, negative 4. What is the value of r and theta? Feel free to pause the video and try. The first equation we need is r. r is equal to the square root of x squared plus y squared. And the second is the angle. The angle is the arc tangent of y divided by x. X, we can see that it's 2, and Y is negative 4. So this is going to be 2 squared plus negative 4 squared. 2 squared is 4, negative 4 squared is 16, and 4 plus 16 is 12. Now we can simplify root 20 if we break it up into 4 and 5. The square root of 4 is 2. So the radius is 2 root 5. So it's going to be the arc tangent. Now let's not worry about the negative sign. We're going to use positive 4 divided by positive 2. Ignore the negative sign initially. So you may need to use your calculator for this problem. Arc tan of 4 divided by 2 is about 63.4 degrees. Now that's the reference angle. In what quadrant is 2, negative 4 located in? 2, negative 4 is located in quadrant 4. X is positive 2, so we've got to travel 2 units to the right. And Y is negative 4. So here it is. The radius is 2 root 5. This angle is the reference angle inside the triangle, which is 63.43 degrees. Now the angle figure that we need is measured from the positive x-axis. So that's 360 minus 63.43, which is about 296.56 degrees. So the answer is 2 root 5 comma 296.56. Let's try this one. Negative 5, 5 root 3. Let's convert it into its polar form. So let's start by finding the radius, which is the square root of x squared plus y squared. So x is negative 5, and y is 5 root 3 squared. Negative 5 squared, that's 25. But what's 5 root 3 squared? 5 root 3 times 5 root 3. If we multiply 5 times 5, that's 25. The square root of 3 times the square root of 3 is 9, and the square root of 9 is 3. 25 times 3 is 75, and 25 plus 75 is 100, and the square root of 100 is 10. So r is equal to 10. So now we need to find the angle theta. So let's use arc tangent 5 root 3 over 5. So keep in mind, theta is r tangent y divided by x. But initially, ignore the negative sign. Let's get the reference angle first. The 5s cancel. So we're looking for the r tan of root 3, which will give us a reference angle of 60 degrees. Now let's find out what quadrant our answer is located in. x is negative and y is positive. So negative 5 is towards the left and positive 5 root 3 is above the x-axis. So the answer is in quadrant 2. And the radius, the hypotenuse of the triangle, is 10. The reference angle is 60, that's the angle inside the triangle. So therefore this angle must be 180 minus 60 which is 120 and that's the angle that we want measured from the positive x-axis. So our answer is 10 comma 120 degrees. And if you want to, you could convert 120 into radians. So this becomes 12 over 18. and 12 is basically 6 times 2, 18 is 6 times 3, so 120 is 2 pi over 3. So you can write your answer as 10 comma 2 pi over 3.

and polar coordinates. What do you think the difference is? Now you're familiar with rectangular coordinates.

Basically it contains an x and y variable, whereas a polar coordinate contains r and theta, where r is the radius of a circle and theta is the angle measured from the positive x axis. So let's say if we wish to plot 3 comma 45 degrees. How can we do so?

So r is 3 and the angle theta is 45. So let's make a graph. The first thing we should do is plot three circles. The first one is going to have a radius of 1, the second one is going to have a radius of 2, and the third one is going to have a radius of 3. So, we're going to draw the ray at a 45 degree angle, and it's going to go to the third circle, because it has a radius of 3. Now, keep in mind, a positive angle is measured from the x-axis rotating counterclockwise.

So that angle is 45. And that's how you can plot a polar coordinate. Now, let's say if we wish to plot the point 2, 3 pi over 4. How can we do so? Now, if you're not sure where 3 pi over 4 is located, convert it to degrees. Let's multiply this by 180 divided by pi.

The pi values will cancel. 180 divided by 4 is 45, and 45 times 3 is 135. So that point is equivalent to 2 comma 135. Now we know this is 0 degrees, this is 90, 180, and 270. So 135 is somewhere in quadrant 2. This is quadrant 1, quadrant 3, and quadrant 4. So we need to have an angle of 135, which is between 90 and 180, and we need to stop at the second circle, because the radius is 2. So this angle here, measured from the positive x-axis, that's 135 degrees, or simply 3 pi over 4. Now what about if r is negative? Let's say if we have negative 2 comma 60 degrees.

How can we plot this particular point? So first, let's draw the circles. This is going to have a radius of 1, a radius of 2, and let's do one more with a radius of 3. Now we know that 60 is between 0 and 90, that's in the first quadrant. And let's plot 2, 60 first. And then we'll plot negative 260 so you can see the difference.

So with a radius of 2, we need to stop at the second circle. And 60 is closer to 90 than it is to 0. So that's where this point is located. Now, if it's negative 2, you need to travel in the other direction. So, you need to travel over here. Notice that negative 2 is the same as positive 2 and 240. So, basically, you got to add 180 to this point.

So, this angle here, that is 60. And this angle here is 60 plus 180, which is 240. So anytime r is negative, instead of going in the direction it should go, go 180 degrees in a direction opposite to where you should go. Let's try another example. Let's use...

let's plot this point, negative 3, 120 degrees. So first, plot 3, 120. So that's going to be in quadrant 2 with a radius of 3. Now let's plot the other one, negative 3, 120. So we're going to travel directly in the opposite direction. So, it should be somewhere over there. So, if this angle is 120, 120 plus 80 is 300, so the other angle is 300, and this angle is negative 60. Sometimes, you may need to represent a point using other values.

So, for example, negative 3 comma 120 is the same as... Positive 3, negative 60. If we had a negative angle, we would go in a clockwise direction as opposed to the counterclockwise direction. And it would lead us to the same point.

Another way in which we can get the answer, we could say that it's also negative 3, 240. So instead of going 120 in this direction, we can travel negative 240 in this direction to get to this point. And because r is negative, it's going to flip back to this point. So to get 240, simply, this is negative 240 by the way, subtract 120 by 360, and that gives you negative 240. Now, we can get another positive r-value, which leads to the same point, if we add 360 to negative 60. So, it's going to be 3,300. So, instead of traveling negative 60 in this direction, we could travel 300 in this direction, also represented by the blue line.

So sometimes you might be given an rtheta value, but you've got to find three other values that would lead to the same point. Two of them are negative, and two are positive. But typically, you might be given one out of those four options. So you'll be given one answer, and you've got to find the other three.

Now let's say if we have the point 2, 30 degrees. And we wish to find the other three points that lead to the same terminal point. And the angle theta has restrictions.

It's between negative 360 and positive 360. Find the other three polar coordinates. So the first one is r theta. The second one will still have a positive r value, but a negative angle. The third one will have a negative r value and a positive angle.

And the fourth will have a negative r value and a negative angle. So we got the first one. It's 2, 30. To find the second one, r is going to be the same. But we're going to subtract 30 by 360. So that's going to be negative 330. So that gives us the second one.

Now for the other two, r has to be negative. So we're going to make it negative 2. And what we need to do is we need to add 180 to our original angle. So 30 plus 180, that's 210. Now to find the other answer, we can either add 180 to negative 330, which is negative 150. Or, we can subtract 210 by 360, which will also give us negative 150. So, this one corresponds to that answer, where both r and theta are negative. And this corresponds to this one, where r is negative and theta is positive.

Now, let's go ahead and plot the original point. So, here's the first circle, and here is the second one. So let's plot 230. So at an angle of 30, the ray has to stop at the second circle.

So this is positive 30 degrees. Now, we can choose to travel in the other direction, and that would be negative 330, which is this answer. Now let's say if we want to use this point.

Positive 210 ends right here. Now keep in mind, we're not going to use this ray because since r is negative, it's going to flip to this point, which is what we want. So that's negative, that's positive 210. Now for the other one, it's negative 150, which will lead us to the same point.

And because r is negative, it flips back to the original terminal point. So that's how you can find the four polar coordinates that lead to the same terminal point. Try this example. Let's say r is 3. and the angle theta is 5 pi over 6. Take a minute and go ahead and find the other three values, the other three polar coordinates that lead to the same terminal point given that the angle theta is between negative 2 pi and 2 pi.

So to find the first point what we need to do is we need to subtract 5 pi over 6 by 2 pi to get a negative angle with a positive r value. Now 2 pi is the same as 12 pi over 6, and 5 minus 12 is negative 7, so we have negative 7 pi over 6. So therefore, that's the second angle. Let's find the next polar point, but first, let's draw it.

So this is the second circle, and this is the third circle. 5 pi over 6 is in the second... quadrant.

It's over there. If you convert it to, let's say, an angle in theta, it's going to be 180 divided by 6, which is 30, because 18 divided by 6 is 3, add the 0, and then 30 times 5 is 150. So this is 5 pi over 6 traveling in that direction. And if we travel in this direction, we still have the same positive r value, but the angle is now negative 7 pi over 6. Now... Let's change 3 into negative 3. If we do that, we need to add pi to 5 pi over 6. So 5 pi over 6 plus pi is the same as 6 pi over 6, so that will give us 11 pi over 6. To get the other negative angle, let's subtract this one by 2 pi.

So 2 pi is the same as 12 pi over 6. And 11 minus 12 is negative 1 pi over 6, or just negative pi over 6. So keep in mind, we need to get to this point, because when the angle is negative, I mean when r is negative, it's going to flip to the red line. 11 pi over 6 is in the fourth quadrant. So it takes us to this point, but because r is negative, it's going to switch to the red line. Negative pi over 6 will take us to the same initial point, and then the negative r will switch us back to the point where we want to be, which is here. So that's how you can find the four polar points given one of the points.

Personally, I think this is helpful. If you know that it's going to be r theta, r negative theta, negative r theta, and negative r negative theta. So, let's say if you're given this point.

To find the second point, simply, you're either adding or subtracting 2 pi. In this case, you're going to subtract it by 2 pi or 360. Since you want to get the negative angle. Now, from number 1 to number 3, you should either add or subtract by pi, or 180. So, if this angle is like 300, subtract it by 180. If it's like 60, add 180 to it. So, if it's more than 180, subtract it by 180. If it's less than 180, add 180 to it. Now, to find number 4, you can just find it from number 3. You can add or subtract by 2 pi.

In this case, subtract by 2 pi since this is positive and you want the negative angle. Now, the next thing we need to go over is how to convert polar coordinates into rectangular coordinates. So you need to know that x is equal to r cosine theta and y is equal to r sine theta. So using those two equations, let's say if we have the point 4 comma 60 degrees.

What is the value of x and y? So x is r cosine theta. So that's 4 cosine 60. And 60 degrees...

is 1 half. So 4 times 1 half is 2. So the x coordinate is 2. Why is r sine theta? So that's 4 sine 60. Sine 60 is the square root of 3 over 2. So 4 divided by 2 is 2, and so it's 2 root 3. And that's how you can convert polar coordinates. into rectangular coordinates.

Try this example 6,5pi over 6. Convert it into rectangular coordinates. So x which is r cosine theta that's 6 cosine 5 pi divided by 6 so what is cosine 5 pi over 6 keep in mind 5 pi over 6 is 150 anytime you see pi over 6 is 30 so 5 pi over 6 is 5 times 30 which is 150 cosine 150 is negative root 3 divided by 2 And 6 divided by 2 is 3, so this is negative 3 root 3. So now we've got to find the y-coordinate, which is 6 sine 5 pi over 6. Sine 150 is positive 1 half. And 6 times 1 half is positive 3. So the answer is going to be negative 3 root 3 comma 3. Now, what if we have rectangular coordinates?

How can we find the value of r and theta? So, for example, let's say if we have the points 2, negative 4. What is the value of r and theta? Feel free to pause the video and try.

The first equation we need is r. r is equal to the square root of x squared plus y squared. And the second is the angle. The angle is the arc tangent of y divided by x.

X, we can see that it's 2, and Y is negative 4. So this is going to be 2 squared plus negative 4 squared. 2 squared is 4, negative 4 squared is 16, and 4 plus 16 is 12. Now we can simplify root 20 if we break it up into 4 and 5. The square root of 4 is 2. So the radius is 2 root 5. So it's going to be the arc tangent. Now let's not worry about the negative sign. We're going to use positive 4 divided by positive 2. Ignore the negative sign initially.

So you may need to use your calculator for this problem. Arc tan of 4 divided by 2 is about 63.4 degrees. Now that's the reference angle.

In what quadrant is 2, negative 4 located in? 2, negative 4 is located in quadrant 4. X is positive 2, so we've got to travel 2 units to the right. And Y is negative 4. So here it is. The radius is 2 root 5. This angle is the reference angle inside the triangle, which is 63.43 degrees. Now the angle figure that we need is measured from the positive x-axis.

So that's 360 minus 63.43, which is about 296.56 degrees. So the answer is 2 root 5 comma 296.56. Let's try this one. Negative 5, 5 root 3. Let's convert it into its polar form. So let's start by finding the radius, which is the square root of x squared plus y squared.

So x is negative 5, and y is 5 root 3 squared. Negative 5 squared, that's 25. But what's 5 root 3 squared? 5 root 3 times 5 root 3. If we multiply 5 times 5, that's 25. The square root of 3 times the square root of 3 is 9, and the square root of 9 is 3. 25 times 3 is 75, and 25 plus 75 is 100, and the square root of 100 is 10. So r is equal to 10. So now we need to find the angle theta. So let's use arc tangent 5 root 3 over 5. So keep in mind, theta is r tangent y divided by x.

But initially, ignore the negative sign. Let's get the reference angle first. The 5s cancel.

So we're looking for the r tan of root 3, which will give us a reference angle of 60 degrees. Now let's find out what quadrant our answer is located in. x is negative and y is positive. So negative 5 is towards the left and positive 5 root 3 is above the x-axis. So the answer is in quadrant 2. And the radius, the hypotenuse of the triangle, is 10. The reference angle is 60, that's the angle inside the triangle.

So therefore this angle must be 180 minus 60 which is 120 and that's the angle that we want measured from the positive x-axis. So our answer is 10 comma 120 degrees. And if you want to, you could convert 120 into radians.

So this becomes 12 over 18. and 12 is basically 6 times 2, 18 is 6 times 3, so 120 is 2 pi over 3. So you can write your answer as 10 comma 2 pi over 3.

Transcript for:Understanding Polar Coordinates and Conversion

Transcript for:
Understanding Polar Coordinates and Conversion