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

Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

Accepted Answer

Welcome back, everyone in this problem. We want to convert the card in its 0 13 that are in rectangular form to pull our card in it. And we want to write our angling radiant for our answer choices. A is the coordinate skirt of 13.5 of pie. B is 13.5 of pi C is a skirt of 13 and three halves of pie and D is 13 and three halves of pie. Now, what, what are we trying to do here? Well, essentially what we're saying is that we have some rectangular coordinates. So we have the coordinates of a complex number uh 0 13. So we can think of it on a grid where the X axis is the real number and the Y axis is the imaginary number. I know we want to plug that complex number and another type of grid. A polar grid that you know, generally looks something like this. So no, our coordinate is going to be in polar form where we'll have our access for the coefficient of the polar number R and or an the on what would have been the Y axis. So if we're going to plot it on a polar grid, polar coordinates, we need it in polar form. So first we can convert our complex number to polar form and then take the information that we need. What do we know about that? Well, recall that if we have a value a complex number in rectangular coordinates. So the form xy, then in polar form, we have it as Z of R and theta where R is the square root of X squared plus Y squared and theta is the arc tangent of Y divided by X. Now, the good thing is we already know the values of X and Y because we know that the real number X is zero and our imaginary part Y is 13. So we can use that to help us solve for R and theta, let's start with R. So this tells us that R will be equal to the square root of X squared and zero squared plus 13 squared, zero squared is zero. So I can rewrite this as 13 squared and the square root of 13 squared that would just be 13. So R equals 13. Next, the would be equal to the inverse, the arc tangent of 13 divided by zero. But if you divide anything by zero, you're going to get an undefined value. So that means our anger or our anger is undefined. No, that would look like it's bad, but it's not really because remember that there are some angles for which the value of the tangent is undefined. Because remember that the tangent of an anger is undefined when our anger equals half of pi and our angle equals three halves of pie. So those are the two possibilities. So we know that our angle is going to be one of them. But again, the question is which one? Well, let's go back to a little sketch that will really help because on our sketch, remember that our point was on the positive y axis. So it follows that in polar, in polar form, it's going to be the same thing. It's also going to be on the positive Y axis. And for our positive y axis, you know, we're just doing so great with the sketches today. Let's come with another one. A half of pi would be on the positive y axis while three halves of pi would have been on the negative y axis. Therefore, we would be using a half of pi. So our value theta this tells us that theta equals a half of pie. Therefore, our coordinates or coordinates would be 13.5 of my. And when we look on our answer choices, that would be answer choice. B 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. (5, 0)

Key Concepts

Rectangular Coordinates

Polar Coordinates

Conversion Between Rectangular and Polar Coordinates

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