Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

Question

Quizplus · Accepted Answer

The polar equation given is in the form of a rose curve with 5 petals, which corresponds to the equation $r = 4\cos(\frac{5	heta}{2})$. To graph this in rectangular form, we can use the conversion formulas $r^2 = x^2 + y^2$ and $x = r\cos	heta, y = r\sin	heta$, and substitute for $r$:
$$(4\cos(\frac{5	heta}{2}))^2 = x^2 + y^2$$
Simplifying, we get:
$$16(\cos^2\frac{5	heta}{2}) = x^2 + y^2$$
$$16(\frac{1+\cos 5	heta}{2}) = x^2 + y^2$$
$$8(1+\cos 5	heta) = x^2 + y^2$$
This is the equation of a cardioid, with the point at the cusp located at $(x,y) = (4,-4)$, which matches the given graph. Therefore, the best choice is C.

Sketch the Graph of the Polar Equation