19.2: Tangent to the Cycloid

The slope of the tangent to the cycloid at P is $dy/dx$, which is equal to $dy/d\theta$, and these can be obtained from Equations 19.1.1 and 19.1.2.

Exercise $\PageIndex{1}$

Show that the slope of the tangent at P is tan $\theta$. That is to say, the tangent at P makes an angle $\theta$ with the horizontal.

Having done that, now consider the following:

Let A be the lowest point of the circle. The angle $\psi$ that AP makes with the horizontal is given by $\tan \psi = \frac{y}{x - 2 a \theta }$

Exercise $\PageIndex{2}$

Show that $\psi = \theta$. Therefore the line AP is the tangent to the cycloid at P; or the tangent at P is the line AP.