Find the highest point on the cardioid $r = 1 + \cos(\theta)$

Question

I'm stuck on this. I don't know where to start!

The problem:

Find the highest point on the cardioid $r = 1 + \cos(\theta)$

Answer 1

As Simon S said, use $y = r\sin \theta$ to get $y = (1 + \cos\theta)\sin\theta$. Then set $dy/d\theta=0$ to maximize $y$: $$ \frac{dy}{d\theta} = (1+\cos\theta)\cos\theta - \sin^2\theta = 0 $$ Use $\sin^2\theta + \cos^2\theta = 1$ to write the equation above as: $$ 2\cos^2\theta + \cos\theta - 1 = 0 $$ The possible solutions are $\cos\theta = 1/2$ and $\cos\theta = -1$. Inspection shows that $\theta = \pi/3$ (or $60º$) is the correct answer.

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