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 contains an X and Y variable whereas a polar coordinate contains R and Theta where R is the radi of a circle Anda is the angle measured from the positive xais so let's say if we wish to plot 3 comma 45° how can we do so so R is three 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's going to have a radius of one the second one is going to 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 Dee 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 xaxis 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 comma 3 piun 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 / Pi the pi values will cancel 180 / 4 is 45 and 45 * 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 one quadrant 3 and Quadrant 4 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° or simply uh 3 Pi 4 now what about if R is negative let's say if we have -2 comma 60° how can we plot this particular point so first let's draw the circles this is is going to have a radius of one a radius of two and let's do one more with a radius of three now we know that 60 is between 0 and 90 that's in the first quadrant and let's plot 2 comma 60 first and then we'll plot -260 so you can see the difference so with a radius of two we need to stop at the second Circle and 60 is closest to 90 then there is a zero so that's where this point is located now if it's -2 you need to travel in the other direction so you need to travel over here notice that -2 is the same as positive2 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 + 180 which is uh 2 240 so anytime R is negative instead of going in the direction it should go go 180° in a direction opposite to where you should go let's try another example let's use let's plot this point -3 comma 120 degrees so first plot 3 120 so that's going to be in quadrant 2 with a radius of three now let's plot the other one -310 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 + 80 is 300 so the other angle is 300 and this angle is -60 sometimes you may need to represent a point using other values so for example -3 comma 120 is the same as POS 3 -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 -3 comma 240 so instead of going 120 in this direction direction we can travel -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 -240 now we can get another positive r value which leads to the same point if we add 360 to-60 so it's going to be 3 comma 300 so instead of traveling 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 got to find the other three now let's say if we have the Point 2 comma 30° 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 - 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 uh 2 comma 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 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 -2 and what we need to do is we need to add one 80 to our original angle so 30 + 180 that's 210 now to find the other answer we can either add 180 to -330 which is -150 or we can subtract 210 by 360 which will also give us - 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° now we can choose to travel any other direction and that would be -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 - 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/ 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 -2 pi and 2 pi so to find the first point what we need to do is we need to subtract 5 piun 6 by 2 pi to get a negative angle with a positive r value now 2 piun is the same as 12 piun / 6 and 5 - 12 is -7 so we have -7 Pi / 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 ID 6 which is 30 because 18 ID 6 is three add the zero and then 30 * 5 is 150 so this is 5 pi/ 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 7 pi over 6 now let's change three into ne3 if we do that we need to add pi to 5 piun 6 so 5 piun / 6 + piun is the same as 6 piun 6 so that will give us 11k 6 to get the other negative angle let's subtract this one by 2 pi so 2 pi is the same as 12 pi/ 6 and 11 - 12 is- 1 piun 6 or just piun 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/ 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/ 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 RGA thet R thet and RGA 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 or 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 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 four you can just find it from number three 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 sin Theta so using those two equations let's say if we have the point 4 comma 60° what is the value of X and Y so X is R cosine thet so that's 4 cosine 60 and 60° is 12 so 4 * a half is 2 so the x coordinate is 2 Y is R sin Theta so that's 4 sin 60 sin 60 is the < TK of 3 over 2 so 4 / 2 is 2 and so it's 2 < tk3 and that's how you can convert polar coordinates into rectangular coordinates try this example 6A 5i 6 convert it into rectangular coordinates so X which is R cosine Theta that's 6 cosine 5 piun / 6 so what is cosine 5 piun over 6 keep in mind 5 pi over 6 is 150 anytime you see pi over 6 it's 30 so 5 pi over 6 is 5 * 30 which is 150 cosine 150 is < tk3 / 2 and 6 / 2 is 3 so this is -3 < tk3 so now we got to find the y coordinate which is 6 sin 5i / 6 sin 150 is positive2 and 6 * 12 is pos3 so the answer is going to be -3 < tk3 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 -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 sare < TK of x^2 + y^2 and the second is the angle the angle is the arc tangent of Y / x x we could see that it's 2 and Y is4 so this is going to be 2^ 2 +4 2 2^2 is 442 is 16 and 4 + 16 is 20 now we could simplify root 20 if we break it up into four and five the square root of 4 is 2 so the radius is 2un 5 now we need to find the angle Theta so it's going to be the AR tangent now let's not worry about the negative sign we're going to use pos4 / pos2 ignore the negative sign initially so you may need to use your calculator for this problem AR tan of 4 / 2 is about 63.4 de now that's the reference angle in what quadrant is 24 located in 2 -4 is located in Quadrant 4 x is pos2 so we got to travel two units to the right and Y is4 so here it is the radius is 2un 5 this angle is the reference angle inside the triangle which is 63.4 3° now the angle the that we need is measured from the positive X axxis so that's 360 minus 63.4 which is about 296.55 de so the answer is 2un 5 comma 29656 let's try this one -5 5 < tk3 let's convert it into its Polar form so let's start by finding the radius which is the root of x^2 + y^2 so X is-5 and Y is 5 < tk3 2-5 that's 25 but what's 5 < tk3 2 5 < tk3 * 5 < tk3 if we multiply 5 * 5 that's 25 the < TK of 3 * the < TK of 3 is 9 and the square root of 9 is 3 25 * 3 is 75 and 25 + 75 is 100 and the square OT of 100 is 10 so R is equal to 10 so now we need to find the angle Theta so let's use AR tangent 5 < tk3 over 5 so keep in mind Theta is AR tangent y / X but initially ignore the negative sign let's get the reference angle first the fives cancel so we're looking for the AR tan of < tk3 which will give us a reference angle of 60° now let's find out what quadrant our answer is located in X is negative and Y is positive so5 is towards the left and positive 5 < tk3 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 60 that's the angle inside the triangle so therefore this angle must must be 180- 60 which is 120 and that's the angle that we want measures from the positive xaxis so our answer is 10 comma 120° and if you want to you could convert 120 into radians so this becomes 12 over 18 and 12 is basically 6 * 2 18 is 6 * 3 so 1 20 is 2 piun over 3 so you can write your answer as 10 comma 2 piun over 3