Tangent to Cycloid is Vertical at Cusps

Theorem

The tangent to the cycloid whose locus is given by:

$x = a \paren {\theta - \sin \theta}$

$y = a \paren {1 - \cos \theta}$

is vertical at the cusps.

Proof

From Slope of Tangent to Cycloid, the slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by:

$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$

At the cusps, $\theta = 2 n \pi$ for $n \in \Z$.

Thus at the cusps, the slope of the tangent to $C$ is $\cot n \pi$.

From Shape of Cotangent Function:

$\ds \lim_{\theta \mathop \to n \pi} \cot \theta \to \infty$

Hence the result by definition of vertical tangent line.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid