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. _ (2,−2√3)

Accepted Answer

Hello everybody. I hope you are doing. All right. Today, today we're going to be looking at this question that states convert the following rectangular coordinates to polar coordinates and write the angle and radiance where the rectangular coordinates given is 21 comma negative seven multiplied by the squared of three. Now the is provided are a seven multiplied by the square of two comma five pi divided by three B seven multiplied by the squared of two comma or pi divided by three C 14 multiplied by the squared of three comma 11 pi divided by six and D 14 multiplied by the squared of three comma seven pi divided by six. Now those as your choices are written as order coordinates. So how do we go from rectangular coordinates to polar coordinates? Well, let's recall exactly how each of those are written. So from rectangular two polar is what we will be doing. Now, rectangular coordinates are written as open parentheses. X comma Y closed parentheses. Now when we change this to porter coordinates, we'll have open parentheses. R comma theta closed parentheses where R is a radius and theta is an angle. Now to calculate R we have a formula that states that R is equal to the square root of open parentheses, X squared plus Y squared, closed parentheses. And that X and Y being the rectangular coordinate version now that we have that in store and now that we know what we're going to be doing, let's sketch our points. So we'll draw a Y axis and an ax axis. And we know that we have a positive X and a negative Y. Therefore, we are in quadrant four. So a draw a line from the origin into quadrant four. And we'll label the XX equals and Y equals negative seven multiplied by the square roots of three. And we label our angle beta. Now we see that our point is located in quadrant four and that is something that we should note and will help us later on in our question. Now that we've just sketched our point, let's solve or R. So let's use our formula R is equal to the square roots of and now we plug in. So we have open bracket, open parentheses 21 which was our X post parentheses, squared plus open parentheses, negative seven, multiplied by the square root of three closed parentheses, squared, close bracket. Now, if we simplify that a bit, we'll have that R is equal to the square roots of 441 plus 147. And once we calculate all of that, we'll have that R is equal to 14 multiplied by the square root of three. And that would be our solution for our R. So we already have half of our po po recording. So now we have to solve or theta which is the other half of our coordinate. Now, if we look at our triangle, we see that we have the X value and the Y value, which means we can use tangent. So have tangent of data is equal to Y divided by X. And this is something that we should know already. So we'll have, we'll plug in our Y and we'll plug in our X. So we'll have negative seven multiplied by the squared of three in the numerator divided by 21 in the denominator. Now, if we simplify that we'll have that tangent of data is equal to negative squared of three divided by positive three. Now, what we can do now is recall from the unit circle and from the unit circle, we know that tangent of pi divided by six is equal to positive squared of three divided by three. So we know that pi divided by six will be our reference angle. Now, we know that our actual angle is in quadrant four. So for a theater in quadrants four, we have a rule and that rule states that our actual theta is equal to two pi minus the reference angle, which in our case is pi divided by six. And this will simplify to have 12 pie divided by six minus pi divided by six, which means that theta is equal to 11 pi divided by six, which means that therefore we will have a change from a rectangular recording of 21 comma negative seven multiplied by the squared of three. That will change to a porter coordinate of 14 multiplied by the squared of three, which is our R comma 11 pi divided by six, which is our beta. So that is the change that we will have. And that will function as our answer answer, which is answer choice C So we can see that by using some simple rules of how to switch from rectangular to polar coordinates, we can rewrite the coordinate that we were given. 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√3)

Key Concepts

Rectangular and Polar Coordinate Systems

Conversion Formulas Between Coordinates

Angle Measurement in Radians and Quadrant Considerations

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