Finding Cartesian from Parametric Equations

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Question

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

Accepted Answer

Hello there. Today we are 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. Identify the particle's path by finding a Cartesian equation for it. Graph this equation indicating both the direction of motion and the proportion. Of the graph traced by the particle for the following parametric equations, X is equal to sin of t minus 3, Y is equal to 2 plus cosine of t, 0 is less than or equal to t, and T is less than or equal to pi. OK. So it appears for this particular prompt we are given parametric equations and a range, and we're asked to use this information to help us identify what this specific particle's path will be by solving for a Cartesian equation. And once we've solved for that Cartesian equation, we need to graph it. And once we have our graph, we need to indicate what the direction of motion will be for the particle itself. So now that we know what we're ultimately trying to solve for, let us first come up with a plan that's come up with a plan for how exactly we're going to approach solving this problem. So as we should recall, in order to solve a problem like this, we will need to first isolate the trigometric terms sine of t and cosine of t, and then we will need to recall and use the fundamental identity. That states that sine to the power of 2 t plus cosine to the power of 2 oft is equal to 1. And we will need to use this fundamental identity to help us eliminate the parameter value t. And this will ultimately reveal that the Cartesian shape will be a circle, and in order to determine the portion of the graph and the direction, we will need to evaluate the x and y values at the boundaries of the interval, which will be 0 is less than or equal to t and t is equal to pi. And we need to note an intermediate point. So once again, in order to determine the portion of the graph and its direction, we will need to evaluate the x and y values at the boundaries of this specific interval and at an intermediate point. So considering this overview for how we're going to solve this problem, our first step that we need to take, like we mentioned earlier, is we need to eliminate our parameter. So we are given that X will be equal to sin oft minus 3, and that y will be equal to 2 plus cosine of T. And we will now need to isolate the trigometric functions, so we need to go ahead and write that sin of t is equal to x + 3. And cosine of t is equal to y minus 2. And when we recall and apply our fundamental trig identity, which states that sine to the power of 2 t plus cosine to the power 2 oft is equal to 1, we can go ahead and write that parentheses of x + 3 to 2 plus parentheses of y minus 2 to the power of 2. is equal to 1, and this will be the equation of our circle, which will be centered at parentheses of 3.2, and it will have a radius R that is equal to 1. For our second step, we need to now identify the portion of the graph that will be traced by our particle. So we will need to look at the range of T from, so we're considering the range of T from 0 to pi. Fantastic. So we need to consider that for the condition 0 is less than or equal to T and T is equal to pi, the function sin of T ranges from 0 to 1, and then it will go back to 0. Therefore, as a result, X will be equal to sin of T minus 3, and this will range from 3 to -2, and then it will come back to -3. And during this same interval, we need to consider that cosine a T, so now we're focusing our attention on cosine a T, and cosine a T will range from negative, so it's gonna be ranging from 1 to -1. Fantastic. So, moving right along here, therefore, Y will be equal to 2 plus cosine of T, and this will be ranging from 3 down to 1. And since, once again, since X is always, so X will be always greater than or equal to -3 throughout this interval, that will mean that the particle will ultimately trace a right semicircle shape. So we will be able to visualize that in a moment. So once again, since X is greater than or equal to -3 throughout this specific interval, that means once again our particle will trace a right semicircle shape for our graph. So moving on now to part 3 of our steps of solving. So step 3, part 3 for our solving process. We now need to determine the direction of motion. So in order to determine the direction of motion, we need to consider the following. So we will need to test the start, middle, and end points of our specific interval. So considering the following values for t. So when t is equal to 0, when t is equal to pi divided by 2, and when t is equal to pi, meaning when we plug in a value of 0 in fort, we will find that x is equal to -3. And that y is equal to 3, meaning our point will be -33. And when we plug in pi divided by 2 in fort, we will find that x is equal to -2 and that y is equal to 2, which will give us the 0.-22. And finally, when we plug in a value of pi in fort, we will find that X is equal to -3, and we will find that Y is equal to 1. So that means we'll have our point 3.1. Which considering these points, we will note that the particle will start at the top of our circle and it will move to the right to the furthest edge and ends at the bottom, and we need to note that this will be a clockwise motion essentially. So with that in mind, we need to move on to our 4th and final step, and our 4th and final step is we need to graph our equation. So we will need to plot the right half of our circle, which will be centered at 3.2 in parentheses. And we need to note that our arc will start at 33, and it will end at, so once again we need to note that it will end at parentheses of 31. And it will pass through, so I'll put the word pass, so I'll pass through the 0.22. And important, this is very important, we need to add arrowheads to indicate the direction of motion, so we need to add an arrowhead on our arc to point downwards in the clockwise direction. Because once again, we need to indicate the motion of the particle on our graph. So using these key pieces of information that we've gathered together, let's look at the blank graph that was given to us by the prom itself. So, I've gone ahead and used online graphing software such as Desmos or MATLAB to create our graph for our curve. And as we can see, we have our semicircle, meaning half a circle shape, with our arrows pointing downwards in a clockwise direction. And our curve is represented by a bold black line. We have our Cartesian equation, which is once again parentheses of x + 32 plus parentheses of Y minus 22 is equal to 1. And this is 4. X is greater than or equal to 3, and as we can see, our particle traces the right semicircle in a clockwise direction, and we have our points, which starts at the beginning parentheses 33 to 22, to finally 31. And that's it. That is our final answer. Hooray, we did it. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. bye.

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