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 one the second one is gonna have a radius of two and the third one is going to have a radius of three 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 three now keep in mind a positive angle is measured from the x-axis rotating counter-clockwise so that angle is forty-five and that's how you can plot a polar coordinate now let's say if we wish to plot the point two comma three pi over four 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 two this is quadrant one quadrant three and quadrant four so we need to go towards we need to have an angle of 1 35 which is between 90 and 180 and we need to stop at the second circle because the radius is two 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 three now we know that sixty is between zero and ninety that's in the first quadrant and let's plot two comma sixty first and then we'll plot negative 260 so you can see the difference so with a radius of two we need to stop at the second circle and 60 is closer to 90 than there's the zero 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 uh 240. so anytime r is negative instead of going in the direction he 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 comma 120 degrees so first plot 3 120. so that's going to be in quadrant two with a radius of three now let's plot the other one negative three 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 sixty sometimes you may need to represent a point using other values so for example negative three 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 comma 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 comma 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 r theta value but you 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 gotta find the other three now let's say if we have the point two comma 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 would have a negative r value and a negative angle so we got the first one it's a two comma thirty to find the second one r is going to be the same but we're gonna 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 gonna make it negative two 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 30 the rate 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 a 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 leads the same terminal point try this example let's say r is 3 and the angle theta is five pi over six 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 two 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 five pi over six 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 seven pi over six now let's change 3 into negative 3. if we do that we need to add pi to 5 pi over 6. so five pi over six plus pi is the same as six pi over six so that will give us 11 pi over six to get the other negative angle let's subtract this one by two pi so two pi is the same as twelve pi over six and eleven minus twelve is negative one pi over six or just negative pi over six 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 that 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 a second point simply you're either adding or subtracting 2 pi in this case you're going to subtract it by 2 pi over 360 since you want to get the negative angle now from number one to number three you should either add or subtract by pi or 180. so this angle is like 300 subtracted 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 four you can just find it from number three you can add or subtract by two pi in this case subtract by two pi since this is positive and you want the negative angle now the next thing we need to go over is how to convert 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 one half so four times a half is two so the x coordinate is two y 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 comma 5 pi over 6 convert it into rectangular coordinates so x which is r cosine theta that's six cosine five pi divided by six so what is cosine five pi over six keep in mind five pi over six is one fifty 150 anytime you see pi over 6 it's 30 so 5 pi over 6 is 5 times 30 which is 150 cosine 150 is negative root 3 divided by 2. and six divided by two is three so this is negative three root three so now we gotta find the y coordinate which is six sine five pi over six sine 150 is positive one-half and six times one-half is positive three so the answer is going to be negative three root three comma three 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 two negative four 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 could see that it's two and y is negative four so this is going to be two squared plus negative four squared two squared is four negative four squared is 16 and 4 plus 16 is 20. now we can simplify root 20 if we break it up into four and five the square root of four is two so the radius is two root 5. now we need to find the angle theta 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 four x is positive two so we gotta travel two units to the right and y is negative four so here it is the radius is two root five this angle is the reference angle inside the triangle which is 63.43 degrees now the angle theta 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 arctangent 5 root 3 over 5. so keep in mind theta is our tangent y divided by x but initially ignore the negative sign let's get the reference angle first the 5's cancel so we're looking for the arc 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 two and the radius the hypotenuse of the triangle is ten the reference angle is sixty as 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 two pi over three so you can write your answer as ten comma two pi over three you