19.2: Tangent to the Cycloid
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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.
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 }
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.