Step By Step Calculus » 14.1 - Polar Co-ordinates and Polar Curves

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Synopsis
In the polar co-ordinate system, every point in the plane is described with respect to a fixed point called pole. The polar axis is drawn from the pole horizontally to the right. Each point PP in the plane can be associated with a pair (r,\theta)(r,\theta) where rr is the straight-line distance from the pole OO and \theta\theta is the angle between the line OPOP and the polar axis.

One can establish a relationship between the Cartesian co-ordinates and the polar co-ordinates as illustrated here.
\[ \begin{array}{ccc} \begin{array}{|c|}\hline \textrm{Polar to Cartesian}\\\hline\\ \left. \begin{array}{l} \cos\theta=\frac{x}{r}\\ \\ \sin\theta=\frac{y}{r} \end{array} \right\}\implies \begin{array}{l} x=r\cos\theta\\ \\y=r\sin\theta. \end{array}\\ \\\hline \end{array} & & \begin{array}{|c|}\hline \textrm{\qquad\qquad Cartesian to Polar\qquad\qquad}\\\hline \begin{array}{l}\\ r=\sqrt{x^2+y^2}\\ \\ \theta=\arctan(x,y)\\ \end{array} \\ \\\hline \end{array} \end{array} \]
A polar curve represents all points in the plane that satisfy a polar equation of the form r=f(\theta)r=f(\theta) or g(r,\theta)=ag(r,\theta)=a.
To find the slope of tangents to a polar curve r=f(\theta)r=f(\theta), we will consider the polar curve in the following parametric form.

\displaystyle { \textbf{(Paramteric Form of Polar Curve)}\qquad \left\{\begin{array}{l} x=r\cos(\theta)=f(\theta)\cos(\theta)\\ y=r\sin(\theta)=f(\theta)\sin(\theta) \end{array}\right\} }\displaystyle { \textbf{(Paramteric Form of Polar Curve)}\qquad \left\{\begin{array}{l} x=r\cos(\theta)=f(\theta)\cos(\theta)\\ y=r\sin(\theta)=f(\theta)\sin(\theta) \end{array}\right\} }


Then, by the equations seen in the review

\displaystyle { \textbf{(Slope of Tangents to Polar Curve)}\qquad \frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{f'(\theta)\sin\theta+f(\theta)\cos\theta}{f'(\theta)\cos\theta-f(\theta)\sin\theta}. }\displaystyle { \textbf{(Slope of Tangents to Polar Curve)}\qquad \frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{f'(\theta)\sin\theta+f(\theta)\cos\theta}{f'(\theta)\cos\theta-f(\theta)\sin\theta}. }