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.

(−2, 2)

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this question as states convert the following rectangular coordinates to polar coordinates and write the angle and radiance where the rectangular coordinates given is negative eight comma positive eight. Now the answer choices provided all provide different versions of the polar coordinate. With answer choice A being eight multiplied by the squared of two comma pi divided by four as a choice B is eight multiplied by the squared of two comma three pi divided by four as a choice C is four multiplied by the squared of two comma five pi divided by four. And as choice D is four multiplied by the squared of two comma pi divided by three. Now, when it comes to a question like this, the first thing we wanna do is well, since we're switching between two different versions of coordinates, how exactly do we do that? Well, let's think about rectangular coordinates and the switch we're going to be making. So we're going from rectangular to poder. Now, how do we do that? Well, rectangular recordings are written as open parentheses. X comma Y closed parentheses. And when we change that to poder, it'll be written as open parentheses. R comma the close parentheses where the R is a radius and the data is an angle. Now, we also have a rule that the R is calculated by having R is equal to the square root of the square root of open parentheses. X squared plus Y squared, close parentheses. Now, when it comes to our point, we should sketch it just to see what it's going to look like. So let's sketch our points. And for that, we draw a Y axis and then X axis and then we'll have a negative X and a positive Y. So we have a point in the second quadrant quadrate number two. And now we have that X is equal to negative eight and Y is equal to positive eight, then we label our angle theta and we have a sketch of our point now that we know what that looks like this will come in handy later on in our question. And I'll show you how now we can move on to solve four R and we just use the formula that we previously mentioned for that. So we have that R is equal to the square root of open brackets, open parentheses, negative eight, closed parentheses, squared plus open parentheses, eight closed parentheses squared, close bracket. Now, if we simplify that a bit further, we'll have that R is equal to the screwed of 64 plus 64. And all I did there was apply the exponents. And if we solve that, now we'll have that R is equal to eight multiplied by the square root of two. So that will be our value for R. And now we have half of our porter cord in it. Now we're missing data. So I think exactly what we'll do. Now we'll solve four the, now we have an X value and A Y value in our sketch. And we know that tangents of theta is equal to Y divided by X, which if we plug the values in, we'll have that, that's equal to eight divided by negative eight. So we'll have that tangent of a theta is equal to negative one. Now, using this information, we can recall from the unit circle, different unit circle values that we should have memorized. So if we recall from the unit circle, we know that tangents of pi divided by four is equal to one. Now we said that our point is in quadrant two. So we know that pi divided by four will be a reference angle. Now, for the in quadrants two, we have a way of solving for the actual angle. So we have the reference angle. Now we're trying to solve for the actual angle. Now for a theta and quad and two, we have that theta is equal to pi minus the reference angle which is pi divided by four. In this case, and that can be rewritten as four pi divided by four minus pi divided by four. And we'll have that theta is equal to three pi divided by four, which means that therefore, we will have a change from a rectangular coordinate of negative eight comma eight to a porter porter coordinate of eight multiplied by the squared of two comma three pi divided by four. And that will be the answer for our question. So we see that it matches with answer choice B and that is the answer that we will select. And we can see how, just knowing how to write polar coordinates and rectangular coordinates. As well as knowing the formula on how to solve for the radius. We know how simple it could be to switch between those two as coins to solve questions just like this one. So I hope that was helpful. And until next time.

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−2, 2)

Key Concepts

Rectangular Coordinates

Polar Coordinates

Angle Measurement in Radians

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