In Exercises 41–48, the rectangular coordinates of a point are gi...

Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians.     _(−√3,−1)

Accepted Answer

Welcome back everyone in this problem. We want to convert the card in it negative 10 and negative multiplied by the spirit of three from rectangular to a polar coordinate. And we want to write the angle answer in radiance for our answer choices A is the coordinate 10 multiplied by the spirit of two and four thirds of pi B is 10 multiplied by the square root of two and five thirds of pi C is 22 3rd of pi and D is 24 3rd of pie. No. What, what are we trying to do here? Well, we're trying to take a value that was in rectangular form. Let me do a little sketch. OK? So it's in rectangular form and it is negative and negative 10 square of three. So it's probably gonna be somewhere down here. OK? Negative 10 and negative multiplied by the square of three. And we're trying to take that point from a rectangular grid and turn it into a point on a polar grid. So we want it eventually probably to look something like this, right? There's gonna be some point in the same quadrant. Now how are we gonna do that? Well, that means first, we'll have to change it to polar farm and then use the polar coordinates. Recall that for a complex number that is uh in rectangular coordinates. So it's in the form X and Y where X is the real number and Y is the imaginary number. Then in polar form, our number would now be with the coordinates R and theta where are equals the square root of X squared plus Y squared. And theater equals the arc tangent of Y divided by X. No. The good thing is we have those values of X and Y. We know that X is negative 10 and Y is negative 10 multiplied by the square of three. So we can substitute those to solve for R and theta respectively. So starting with R, this tells us that R equals the square root of X squared that has negative 10 R squared plus negative 10 multiplied by the square of three R squared. No, when we were old negative 10 squared would be negative 10 multiplied by the square of three, all squared would have been negative 10 squared, which is 100 and the square would have three squared, which is three. So that would be 100 times three, which would have been 300 100 plus 300 is 400 and the square root of is 20. So R equals 20. Next, we need to find our angle. So theta would be equal to the arc tangent of Y divided by X that negative 10 multiplied by the square of three divided by negative 10. Now the negative tense cancel. And that means our anger would be the inverse tangent of the square root of three. In other words, what value of theater gives us the square root of three? Well, if we recall on our unit circle, OK, then there are two values for that. So our angle could either be a third of pi or it could have been four thirds of pie. So the question is, which one are we going to use? Well, let's go back to our sketch, right? Remember that for our value in a rectangular cord in it, it was in the third quadrant. OK? And in the third quadrant would wear the four thirds of pi. In other words, if I were to put that on my rectangular cord, my rectangular grid, then that is where four thirds of pie would be. That means the value for theater that we're going to use is four thirds of pie. So since theta is in quadrant three, then we're going to let theater equal four thirds of pie. Therefore, or polar coordinates. OK? Would be R which is theta which is four thirds of pi. And if we go back to our answer choices, that would have been answer choice. D Thanks a lot for watching everyone. I hope this video helped.

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _(−√3,−1)

Key Concepts

Rectangular Coordinates

Polar Coordinates

Angle Measurement in Radians

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