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 = 1 + cos t , y = sin t ― 1

Accepted Answer

Welcome back, everyone. Given the parameteric equations X equals 4 cosine T minus 4 and Y equals 4 T + 44 TN 0 to TI inclusive. Grab the corresponding curve. What is the rectangular equation of this curve? So first of all, we're going to identify the rectangular equation of the curve, and basically we're going to begin with X equals 4, cosine T minus 4. What we can do is basically add 4 to both sides, right? So X plus 4. Equals 4 cosinet. And now we can solve for the value of cosine. Now, if we divide both sides by 4, we end up with X + 4 divided by 4 equals cosine of T. Now from here we're going to look at the equation Y equals 4 of T + 4, and we are going to solve for sin of T. So once again we're subtracting 4 from both sides and dividing by 4. This gives us sine T. And now based on the Pythagorean identity, we know that signs where. Of T plus cosine squared of T. Is equal to one in our case. If we take. X plus 4 divided by 4, and if we square it. This is going to be called sine squared of T, right? And then we're going to take sine squared of T. So essentially we get Y minus 4 divided by 4 squad. Or sine squared of t and this sum must be equal to 1. Now notice that these two have the same denominator, so we can just factor it out, right? We're going to get 1. Divided by 16 in rancis. X + 4 squared. Plus Y minus 4 squad is equal to 1 and now multiplying both sides by 16, we get X plus 4 2 plus y minus +42 equals 16 and this is the equation of a circle, right? It has a center at -4 and 4. Right, because the general equation of a circle is X minus A2 + Y minus B2 equals R2 where R is radius, and the center is at AB. In this case, X minus -4 gives us X + 4, so A is -4 and B is 4. So we want to plot a circle with a center at -4 and 4, and the radius of the circle is 4. We're going to look at the actual graph. Of this circle. And we're going to understand how to plot it. So, first of all, we want to add the center at 4. This is our Y coordinate and -4. For X we want to find the intersection point. And this gives us the center. Now, our radius is 4, so what we want to do is basically extend. A line from the center, 4 units above the center. So we're going to get a value of 8, and if we go down from 4, we get 0, right? So we essentially want to make sure that we get those points, and also we can go to the left and to the right. If we go to the right by 4 units from -4, we end up at The y axis, and if we go to the left, we must get -8 from -4, we're going by 4 units to the left. This gives us -8. So when we have those 4 points, right, we can essentially plot our curve. It is a circle because T is between 0 and 2 pi. So this means complete revolution. And that's it, we have identified the answer to this problem and we have our curve. So first of all, let's recall that the equation is. X + 4. Square plus. Y minus 42 equals 16 and we have our curve. 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 = 1 + cos t , y = sin t ― 1

Key Concepts

Parametric Equations

Graphing Parametric Curves

Rectangular Equation

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