In Exercises 21–40, eliminate the parameter t. Then use the re...

Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.

x = 2 sin t, y = 2 cos t; 0 ≤ t < 2π

x = 2 sin t, y = 2 cos t; 0 ≤ t < 2π

Accepted Answer

Below there. Today we're 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. Write the corresponding rectangular equation for the following parametric equation by eliminating T. Draw a graph of the plane curve using the rectangular equation. Indicate the direction of the curve that is obtained by using arrows that correspond to the increasing values of T. Awesome. So it appears the entire problem itself is all the clues we need to solve this problem, which is awesome. So it appears what we're ultimately trying to solve for is two separate answers. So we're trying to figure out the corresponding rectangular equation for the following parametric equation. And we're asked to do it by eliminating T, and then we're also asked, that's our first answer. Our second answer we're trying to solve for is we're trying to draw a graph of the plane curve using this rectangular equation. And then thirdly, our last answer that we're trying to solve for is we're trying to figure out the direction of the curve that is obtained by using arrows that correspond to the increasing values of T. So on our graph, we're trying to show using arrows what the direction of the curve is, depending on the increasing values of T. So now that we know what we're solving for, Let's take a moment here to look at this parametric equation that's provided to us. So we're told that X equals 4 multiplied by sin of T and Y equals 4 multiplied by cosine of T, where 0 is less than or equal to T and T is less than 2 pi. Awesome. So, with that in mind, now that we know what our parametric equation is, let's read off our multiple choice answers to see what our final answer pair might be. So A is x2 + y2 equals 16 and clockwise, where clockwise denotes the direction. B is X2 + y2 equals 16 counterclockwise. C is X2 + y2 equals 4 clockwise, and finally D is X2 + y2 equals 4 counterclockwise. OK. So first off, we need to consider our parametric equation once again. So X, so when you say that X, when you take X is equal to. Or multiplied by sign of T. And we also need to consider Y equals 4 multiplied by cosine of T. Awesome. And we need to note that our boundaries here are gonna be 0 is greater, so 0 is less than or equal to T and T is less than 2 pi. Awesome. So in order to eliminate the parameter T, as we should recall and note, we need to use trigontric identities for sine and cosine. So if we're given that X equals 4 multiplied by sina T and Y equals 4 multiplied by cosine a T, we need to square both equations. So when we do that, we will find that X2 is equal to 16 multiplied by squared of T. And we'll also find that Y2 is equal to 16 multiplied by cosine squared of T. Awesome. So, moving right along here. So now we need to add these two equations to get, and when we add these two equations you'll get X2 plus Y2 is equal to 16 multiplied by sine squared. Of T plus cosine squared of T. And since sine squared of T plus cosine squared of T is equal to 1, so let's underline this. So the sine squared of T plus cosine square, so sine square to T plus cosine squared of T, this expression is equal to 1. Fantastic. So the equation so therefore our equation will simplify to simply X2 + Y2 equals 16. And this will represent a circle with a radius 4 that is centered at the origin. So this right here is our equation that we solve for. So we solve for our first answer. Hooray, we did it. So we found our corresponding rectangular equation, fantastic. So now, now that we know that this will represent a circle with a radius 4 that is centered at the origin, we can now begin to draw our graph of X2 + Y2 equals 16. So I've gone ahead and used my graphing software to plot our graph. So as we could see, I'll bring it up a little bit. Awesome. So as you can see, We have our table here, so we created a table, so we are using our table to help us draw a graph of X2 + Y2 equals 16 using different points. So in our first column, we have T equals 0, and then we have T equals pi divided by 2, then 3 pi divided by 2 and 2 pi. And this corresponds to X where for when T equals 0, X is 0. When T is pi divided by 2, X is 4, and when T is i, X is 0, and when T is 3 pi divided by 2, X is -4, and when T is 2, X is 0. And similarly, all the values for Y corresponding to these values of T are 40, -4, 0, and 4. And as you can see when we plot these graphs, since we have a radius of 4, so noting once again, and I'll draw in red, we have, it starts, so we have our circle, which is centered at the origin. So the red dot is our origin point, which is the center, and it has a cert so this is a circle of radius 4. And when we plot our different values, you will see that we have our curves that form our circle. And for increasing values of T, we could see that the arrows are pointing to the right and they're going in the clockwise direction. So therefore, our final answer, without a doubt has to be multiple choice answer letter A, which once again is X. Squared plus y2 equals 16 clockwise. And once again, the arrows are pointing into the increasing values of T. So T is increasing in the clockwise direction, as you can see, based off of the information we gather together in our table. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Parametric Equations

Eliminating the Parameter

Graphing the Curve

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