How to find horizontal and vertical tangents to a polar curve

A polar equation describes a curve using radius and angle, but a tangent line still has a direction in the usual coordinate plane. We can find that direction by writing both coordinates in terms of the angle. Then the two derivatives tell us where the curve is moving sideways or straight up and down.

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Find the two coordinate derivatives

Use $x=r\cos \theta$ and $y=r\sin \theta$. A horizontal tangent occurs when $dy/d\theta =0$ and $dx/d\theta \ne 0$. A vertical tangent occurs when $dx/d\theta =0$ and $dy/d\theta \ne 0$. If both vanish, examine the curve near that point before deciding the direction.

Use the conditions on the polar circle

For $r=2\sin \theta$, the circle is traced once as $\theta$ runs from $0$ to $\pi$. The endpoint at $\pi$ repeats the starting point.

$\begin{array}{l}x=\sin \left(2\theta \right)\\ y=1-\cos \left(2\theta \right)\\ \frac{dx}{d\theta }=2\cos \left(2\theta \right)\\ \frac{dy}{d\theta }=2\sin \left(2\theta \right)\end{array}$

The horizontal tangents occur at $\theta =0$ and $\theta =\frac{\pi }{2}$, giving $(0,0)$ and $(0,2)$ in rectangular coordinates. The tangent lines are $y=0$ and $y=2$.

The vertical tangents occur at $\theta =\frac{\pi }{4}$ and $\theta =\frac{3\pi }{4}$. The radius is $\sqrt{2}$ at both angles. The points are $(1,1)$ and $(-1,1)$, and the tangent lines are $x=1$ and $x=-1$.

The circle has center (0, 1) and radius 1.

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