We need to sketch `r = 2.5`

In this example, we cannot see "θ" in the function we are given. This means the radius, r is constant, no matter what value angle θ takes.

Let's investigate this using the following interactive graph.

Drag the blue dot left and right to change the angle θ. Does the size of r change as you change the angle?

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Hint: Let go of the blue dot to see a smooth curve at any time. You can trace out positive or negative angles.

Here's an image in case you can't see the graph above.

30°60°90°120°150°180°210°240°270°300°330°0123

Graph of r = 2.5, a limacon.

What is the Equivalent in Rectangular Coordinates?

We convert the function given in this question to rectangular coordinates to see how much simpler it is when written in polar coordinates.

To convert `r = 2.5` into rectangular coordinates, we use

r2 = x2 + y2

In this example, `r = 2.5`, so `r^2= 6.25`.

So this gives us: x2 + y2 = 6.25

Not surprisingly, this is the similar to the equation for a circle we obtained in The Circle section earlier.