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.