Graph each plane curve defined by the parametric equations for...

Question

Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.

x = 2 + sin t , y = 1 + cos t

Accepted Answer

Welcome back, everyone. Given the parameter equations X equals 1 + 3 st and Y equals -2 + 3 cosine T or T in the interval between 0 and 2 pi inclusive, graph the corresponding curve. What is the rectangular equation of this curve? So first of all, let's identify the rectangular equation. We're going to do that by solving for sine T and cosine of T first. So for sine C, we have to subtract one from both sides and divide by 3. So we got X minus 1 divided by 3. This is the expression of sine C. Now for Y, we're going to solve for cosinet. And essentially we have to add 2 to both hand sides and then divide by 3, which is the leading coefficient in front of cosine T. Now we're going to square them. We want to use. Pythagoan identity, which tells us that sine squared of T plus cosine squared of T is equal to 1. So we're going to get X minus +12 divided by V2. Plus Y + 2 squared divided by 3 squad. Is going to be equal to 9 and is going to be equal to 1. I'm sorry, right, and then we can multiply both sides by 3 squared. This gives us X minus 12 + y + 2 quad is equal to v2 which is 9. And let's recall that this is an equation of a circle. X minus a2 plus Y minus B2 equals R2. And we can tell that our center is at A, B, that's our center. And our radiuses are, so looking at this problem, we have our center at. A which is one. And our B is -2 because we can say that Y + 2 is. Y minus -2. So our B is -2 and our radius is. Square root of 9, right, which is 3, so we want to plot a circle. With the center at 12. Let's suppose we make a rough sketch. X is 1, Y is -2. We can add our center and then our radius is 3, and this basically means that we want to add 4 more points. So first of all, if we go from our center upwards, we want to count 3 units -2 + 3 gives us 1. Then we want to go to the right 1+ hour radius gives us 4. So we have 123, and 4. Essentially we're going to add. Another point, now we go down from negative to. We're going to -5, because we're adding -2 minus 3 since we're going down, right? And then What else? Well, we are going to the left, so we're going from point X equals 13 units to the left, so 1 minus 3 gives us -2. And basically this gives us our circle. We just have to connect those 4 points. Now this is just a rough sketch. Now, let's consider the actual graph using a graphing utility. If we consider the actual graph, we can understand that our center is at 0.12. This is exactly what we have. And then we are simply adding those 4 points. One of them is. Or 2. The other one is at our X of 1. And why of one. Then we have another point at X of -2. And Y of -2. And finally we have our 4th point at X of 1. And our Y value is -5. And when we connect those points, well, we're going to get our circle. That's it. Thank you for watching.

Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.

x = 2 + sin t , y = 1 + cos t

Key Concepts

Parametric Equations

Graphing Techniques

Rectangular Equation

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