In Exercises 81–82, find the rectangular coordinates of each pair...

Question

In Exercises 81–82, find the rectangular coordinates of each pair of points. Then find the distance, in simplified radical form, between the points.

(2, 2π/3) and (4, π/6)

Accepted Answer

Hello, today we're going to be converting the two given coordinates into rectangular form. And we're going to find the distance between the two given points. So one thing to note is that the given coordinates are in polar form. The general form of the polar form of coordinates is given to us as our comma theta. What we wanna do is you want to change our comma theater data into rectangular form. And the general rectangular form of a coordinate is given to us as X comma Y. While we do have two conversion factors to help us go from our comma theta to X comma Y, we are given X is equal to R multiplied by cosine of data. And we are given Y is equal to R multiplied by sine of data. So let's start by converting the first coordinate from polar to rectangular form. The first coordinate is given to us as four comma three pi over four. So let's first get the X term of this coordinate and again, X is equal to R multiplied by cosine of the, in this case, R is equal to four and theta is equal to three pi over four. So what this means is that X is equal to four multiplied by cosine data which will be cosine of three pi over four cosine of three pi over four will evaluate to give us negative square root 2/2. So we have four multiplied by negative square root 2/2. And multiplying these two values together will give us an X value of negative two multiplied by the square root of two. Next, let's get the Y value for this coordinate. Now again, Y is equal to R multiplied by sine of data which will be four multiplied by sine of three pi over four sine of three pi over four will evaluate to give us positive square root 2/2. So we have four multiplied by the positive square root 2/2. And multiplying these values together will give us a final product of two square root of two. So the rectangular form of the first coordinate will have a rectangular form of negative two square root of two comma two square root of two. And for now, we're going to label this as X one and Y one. Next, let's change the second coordinate into po from polar to rectangular form. Now the second coordinate is given to us as eight comma pi over four. In this case, R is going to equal to eight and theta is going to equal to pi over four. So using these values, we can get our X and Y values for the rectangular version of this coordinate X is equal to R multiplied by cosine data which will be eight multiplied by cosine pi over four cosine pi over four will evaluate to give us positive square root 2/2. So we have eight multiplied by square root 2/2. And multiplying these values together will give us a final product of four multiplied by square root of two. Next, we'll get our Y value which is R multiplied by sine theta. And that will be eight multiplied by sine of pi over four sign of pi over four will evaluate to give us square root 2/2. So we have eight multiplied by square root 2/2. And that will simplify to leave us with four square root of two. So what this means is that the rectangular form will be four square root of two comma four square root of two. And we're going to label this point X two and Y two. So what we just did is we converted the first coordinate from rectangular to polar form and we converted the second coordinate from polar to rectangular form. Next, we're going to move on to the second part of the problem which asks us to find the distance between the two points. So recall that the distance formula is given to us as D is equal to the square root of Y two minus Y one squared plus X two minus X one squared. Now earlier, we labeled the two rectangular coordinates, we have Y two which is four square root of two and Y one which is two square root of two. So if we plug this into the distance formula, we have the square root of the quantity four square root of two minus two square root of two, that quantity is squared. And then we're going to be adding X two, which we labeled as four square root of two minus X one, which is negative two square root of two and negative one multiplied by that value will leave us with positive two square root of two. And that quantity is squared. Next, what we're going to do is we're going to simplify these values four square root of two minus two square root of two will give us a final value of two square root of two squared and four square root of two plus two square root of two will give us six square root of two. And that quantity is squared. Next two square root of two squared will simplify to give us four multiplied by two and six square root of two squared will simplify to give us 36 multiplied by two. Next four, multiplied by two will give us the value of eight and 36 multiplied by two will give us the value of 72. Finally, eight plus 72 will give us a final value of the square root of 80 the square root of 80 will simplify to leave us with four multiplied by the square root of five. This is going to be the distance between the two given points. So just to summarize the rectangular form of the first coordinate four comma three pi over four is negative two square root of two comma two square root of two. And the rectangular form of the second given polar coordinate is four square root of two comma four square root of two. And the distance between these two points is going to be four multiplied by the square root of five. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 81–82, find the rectangular coordinates of each pair of points. Then find the distance, in simplified radical form, between the points. (2, 2π/3) and (4, π/6)

Key Concepts

Polar Coordinates

Rectangular Coordinates

Distance Formula

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