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Problem 710

# Graph the curve $$\mathrm{r}=2+2 \cos \theta$$.

Expert verified
To graph the curve $$r = 2 + 2\cos\theta$$, plot critical points such as (4, 0), (2, π/2), (0, π), (2, 3π/2), and (4, 2π) in the polar coordinate system. Connect these points smoothly to form a loop shape known as a limaçon, which has symmetry with respect to the polar axis. The graph should show the curve originating from the point (4,0), extending above and below the polar axis, and closing back at the same point (4,0).
See the step by step solution

## Step 1: Understanding the Polar Coordinate System

In a polar coordinate system, a point is defined by its distance (r) from the origin, and the angle (θ) it makes with the positive x-axis. The relationship between rectangular coordinates (x, y) and polar coordinates (r, θ) is: $x = r\cos(\theta)$ $y = r\sin(\theta)$

## Step 2: Analyzing the Equation

Given the equation $$r = 2 + 2\cos\theta$$, notice that it contains a $$\cos\theta$$ term, which tells us that this curve will have symmetry with respect to the polar axis (θ = 0). Also, the value of r depends on the angle θ, which means that as θ changes, r will also change. To better understand the behavior of the curve, we can analyze the equation at critical points and various other values of θ: - At θ = 0: $$r = 2 + 2\cos(0) = 2 + 2(1) = 4$$ - At θ = π/2: $$r = 2 + 2\cos(\pi/2) = 2 + 2(0) = 2$$ - At θ = π: $$r = 2 + 2\cos(\pi) = 2 + 2(-1) = 0$$ - At θ = 3π/2: $$r = 2 + 2\cos(3\pi/2) = 2 + 2(0) = 2$$ - At θ = 2π: $$r = 2 + 2\cos(2\pi) = 2 + 2(1) = 4$$ This tells us that our graph will have a maximum r value of 4 and a minimum r value of 0.

## Step 3: Plotting the Points

Plot the points from our analysis above: - (4, 0) - (2, π/2) - (0, π) - (2, 3π/2) - (4, 2π)

## Step 4: Connecting the Points and Drawing the Curve

After plotting these points, connect them smoothly to form a continuous curve that has symmetry with respect to the polar axis. Make sure to create a smooth "loop" shape, as the r-values increase to 4 and then decrease to 0 (with θ going from 0 to π), and then from 0 to 2 (with θ going from π to 2π). The final graph should show a loop shape known as a limaçon, originating from the point (4,0) on the polar axis, extending above and below it, and closing back at the same point (4,0). Your graph of the curve $$r = 2 + 2\cos\theta$$ is complete.

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