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

Step 1: Understand the parametric equations given: $$x = 1 + \cos t$$ and $$y = \sin t - 1. $$These equations describe a curve in the plane as the parameter $$t$$ varies from 0 to $$2\pi. $$Step 2: To graph the curve, consider the range of $$t$$ from 0 to $$2\pi. $$For each value of $$t$$, calculate the corresponding $$x$$ and $$y$$ values using the parametric equations, and plot these points on the Cartesian plane. Step 3: To find a rectangular equation, eliminate the parameter $$t. $$Start by expressing $$\cos t$$ and $$\sin t in $$terms of $$x$$ and $$y. $$From $$x = 1 + \cos t$$, we have $$\cos t = x - 1. $$From $$y = \sin t - 1$$, we have $$\sin t = y + 1. $$Step 4: Use the Pythagorean identity $$\cos^2 t + \sin^2 t = 1 to $$combine these expressions: $$(x - 1)^2 + (y + 1)^2 = 1. $$This equation represents a circle centered at $$(1, -1)$$ with a radius of 1. Step 5: Verify the rectangular equation by checking that it satisfies the original parametric equations for various values of $$t$$ within the given range. This ensures that the transformation from parametric to rectangular form is correct.

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

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