The parametric equations of Involute of a circle is and .

Find the points at which the tangent is horizontal.

For Horizontal tangent: .

Consider .

Apply derivative on each side with respect to .

.

Equate to .

and .

and .

Horizontal tangents are  at .

Find the set of points.

At :

.

At , the point is .

At :

.

At , the point is .

At :

.

At , the point is .

Hence points are and , where is a integer.

Horizontal tangent line occur at points , and .

For Vertical tangent: .

Consider .

Apply derivative on each side with respect to .

.

Equate to .

and .

As is not the solution then .

Vertical tangents are at .

Find the set of points.

At :

.

At , the point is .

At :

.

At , the point is .

At :

.

At , the point is .

Hence points are , where is a integer.

Vertical tangent line occur at points , and .

Horizontal tangent line occur at points , and .

Vertical tangent line occur at points , and .