FP2.7 - Polar coordinates

A polar point $(r,\theta)$ lies a distance $r\ge0$ from the pole in the direction measured by $\theta$ from the initial line. Cartesian and polar descriptions are connected by$$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$At the pole $r=0$, the angle is not unique.

To sketch reliably, first state the permitted $\theta$ interval. Test symmetry by replacing $\theta$ with $-\theta$ or $\pi-\theta$, solve $r=0$ for passages through the pole, find maximum and minimum $r$, and calculate a small table at exact angles. Plot in increasing $\theta$ so the order of loops or arcs is clear.

Cartesian conversion can confirm the shape. For example,$$r=2a\cos\theta\quad\Longrightarrow\quad r^2=2ar\cos\theta,$$so $$x^2+y^2=2ax\quad\Longrightarrow\quad (x-a)^2+y^2=a^2.$$This proves that the polar curve is the circle described in the table.

Curves drawn over the same ray intersect where their admissible radii are equal. Solve $r_1(\theta)=r_2(\theta)$, retain angles in the stated range, and substitute back for the exact radius. The pole may be a shared point even when it arises from different angle values.

A narrow polar sector of angle $d\theta$ has area approximately $\tfrac12r^2d\theta$. Therefore the area swept once as $\theta$ runs from $\alpha$ to $\beta$ is$$A=\frac12\int_{\alpha}^{\beta}r^2\,d\theta.$$The limits must describe the actual boundary rays or curve intersections of the required region.

For the upper half of the cardioid $r=a(1+\cos\theta)$, $0\le\theta\le\pi$,$$A=\frac{a^2}{2}\int_0^\pi(1+2\cos\theta+\cos^2\theta)\,d\theta.$$Using $\cos^2\theta=\tfrac12(1+\cos2\theta)$ gives$$A=\frac{a^2}{2}\left(\pi+\frac\pi2\right)=\frac{3\pi a^2}{4}.$$

For $r=2a\cos\theta$, at $\theta=\pi/4$ we have $r=\sqrt2a$. Also$$\frac{dy}{d\theta}=2a\cos2\theta=0,\qquad \frac{dx}{d\theta}=-2a\sin2\theta=-2a\ne0.$$Thus the tangent is parallel to the initial line at $(r,\theta)=(\sqrt2a,\pi/4)$, the top point $(a,a)$ of the circle.