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 cos t , y = 2 sin t

Accepted Answer

Welcome back, everyone. Given the parametric equations X equals 17 cosine C and Y equals 17 st, where T in the interval between 0 and 2 pi inclusive, grab the corresponding plane curve. What is the rectangular equation of this curve? So first of all, let's consider the equation itself. We have X equals 17 cosinet and Y equals 17 sinet. If we solve for cosine and sine, we're going to get cosine of T equals x divided by 17, and sine of T is equal to. Why divided by 17. Now let's make use of Pythagorean identity. We know that signs where. T plus cosine quadt must be equal to 1. So if we square y divided by 17 and then add the square of x divided by 17, we're going to get 1. Now, first of all, we're going to square 17, which is 289, so we get 1 divided by 289, we're just going to factor it out. Multiplied by y squared. Plus 1 divided by 289, which is 1 divided by 17 squad, right, and then we have to multiply it by x2. This must be equal to 1. If we multiply both sides of the equation by 289, we're going to get X2 + y2 equals 289. And notice that this is the equation of a circle which has a center at the origin, right? Our center is at the origin. And then what about the radius? Well, our radius is going to be equal to square root of 289. Because we are considering an equation in the form of X2 + Y2 equals R2. So when we take square of 289, we're going to get 17 for our radius. So we want to plot a circle with The center at divergin and then we want to make sure that the radius is 17, so we can just add. 4 points showing a distance of 17 up. To the right To the left In going down. Going down, our coordinate would be negative, so we're going to label -17 just as we did to the left, right? So those are the four points and then we can just connect them in a circle because T is from 0 to TI inclusive. This means a complete revolution. So what we're going to do is just connect those points in a circle. And that's it, we have our final answer. 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 cos t , y = 2 sin t

Key Concepts

Parametric Equations

Graphing Parametric Curves

Rectangular Equation

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