Question

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

Answer 1

Need to maximize the y-component or \(r * \sin(\theta) = \sin(\theta)(1 + \cos(\theta))\)

Answer 2

now where does rsin(theta) come from?

Answer 3

Polar coordinates to Cartesian coordinates.
x = r * cos(theta)
y = r * sin(theta)

Answer 4

Highest point is when y is maximum.
\(y = r * \sin(\theta) = \sin(\theta)(1 + \cos(\theta))\)
Set \(\large \frac{dy}{d\theta} = 0\) and solve for \(\theta\).
Find which \(\theta\) would give the highest y. Put that \(\theta\) in r and calculate r.