Questions: Sketch the graph of r=3+2 sin theta.

Transcript text: Sketch the graph of $r=3+2 \sin \theta$.

Solution

Solution Steps

Step 1: Identify the type of curve

The equation is in the form $r = a + b\sin(\theta)$. Since $a/b$ (3/2 = 1.5) is between 1 and 2, this is a dimpled limaçon.

Step 2: Determine key points

When $\theta = 0$, $r = 3 + 2\sin(0) = 3$.
When $\theta = \pi/2$, $r = 3 + 2\sin(\pi/2) = 5$.
When $\theta = \pi$, $r = 3 + 2\sin(\pi) = 3$.
When $\theta = 3\pi/2$, $r = 3 + 2\sin(3\pi/2) = 1$.

Step 3: Sketch the curve

Plot the points calculated in Step 2. Knowing that the curve is a dimpled limaçon, connect the points smoothly, creating a dimple on the "bottom" of the curve, where the radius is at its minimum (1). The provided image accurately represents the graph of this equation.

Final Answer: The graph provided in the image is correct.

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