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# 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.