Cartesian to Polar Coordinates

Find the polar...

Question

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.

c. (−1, √3)

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Convert the Cartesian coordinates parentheses of 1, square root of 3 to polar coordinates with 0 is less than or equal to theta and theta is less than 2 pi and r is less than or equal to 0. OK, so it appears for this particular prom we're asked to convert the Cartesian coordinates that are given to us to polar coordinates given the specific conditions laid out to us by the prom itself. So now that we know that we're converting Cartesian coordinates to polar coordinates. We need to recall a note that based on our prior knowledge in order to convert the Cartesian coordinates parentheses of x, Y is equal to 1, square root of 3, and where in order to convert these Cartesian coordinates to polar coordinates, which polar coordinates are written as parentheses of R, theta. And we need to note once again that our conditions are given to us as R is less than or equal to 0 and 0 is less than or equal to theta and theta is less than 2 pi. And in order to perform our conversion, we need to note that our first step that we need to take is we need to find the magnitude of the radius and as we should recall based on our prior knowledge, the distance of the point from the origin is given by the absolute value of r, which is equal to the square root of. X2 + y2, which is equal to the square root of 12 minus it is going to be 12 plus parentheses of square root of 3 to the 2, and this will be simply equal to 2 when we plug that expression into our calculator. And we need to recall a note, since the prom requires that R is less than or equal to 0, we must choose that R is equal to -2. For our second step, we need to determine what the standard angle is for a positive r value. So considering the points of 1 square root of 3, we need to note that this specific point is in the 4th quadrant, which we could write as parentheses of x y. And the standard angle alpha or R is equal to 2 will be written as tangent of alpha is equal toy divided by x, which will be equal to square root of 3 divided by 1, which will be equal to square root of 3. So in the 4th quadrant, so let's make a note that in the 4th quadrant, alpha will be equal to 5 pi divided by 3. Fantastic. So for our 3rd step, so this is step 3, we need to adjust the angle for negative R. So, as we should recall, to represent the same point with a negative radius, we must first shift the angle by pi radiance such that the terminal side points to the opposite direction, so it'll be pointing in the opposite direction. So you can go ahead and write that theta will be equal to alpha minus pi, which will be equal to 5 pi divided by 3 minus pi. Which will be equal to 2 pi divided by 3. Now we need to verify the constraints, so we need to note that R is equal to -2 and 2 is less than or equal to 0, so that means it's satisfied. 0, once again, now we need to consider 0 is less than or equal to 2 pi divided by 3, which is less than. 2 pi and that condition is also satisfied. So these constraints are both satisfied. So moving on to our verification, we need to note that when we use R is equal to -2 and when we use that theta is going to be equal to 2 i divided by 3. We need to note that we can go ahead and write that X will be equal to R multiplied by cosine of theta, which will be equal to -2 multiplied by cosine of 2 pi divided by 3, which will be simply equal to 1. Andy will be equal to r multiplied by sin of theta, which will be equal to -2 multiplied by sin of 2 pi divided by 3, which will be equal to the square root of 3. Thus, the polar coordinates without a doubt will be parentheses of 22 pi divided by 3, and that's it. That is our final answer. Hooray, we did it. So going back to the top to recap what we've learned. We could safely say without a doubt that our final answer for the polar coordinates will be parentheses of 22 pi divided by 3. That's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Cartesian to Polar CoordinatesFind the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.c. (−1, √3)

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Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.

c. (−1, √3)