Chapter 10.3, Problem 62ES

Use the formula for $\psi$ obtained in Exercise 61.

(a) Use the trigonometric identity

$\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\sin \theta }$

to show that if $\left(r,\theta \right)$ is a point on the cardioid $r=1-\cos \theta \left(0\le \theta <2\pi \right)$ then $\psi =\theta /2.$

(b) Sketch the cardioid and show the angle $\psi$ at the points where the cardioid crosses the $y\text{-axis}\text{.}$

(c) Find the angle $\psi$ at the points where the cardioid crosses the $y\text{-axis}\text{.}$