Formulae to convert Cartesian to polar?

r=\sqrt{x^2+y^2}, \tan\theta=\frac yx (check the quadrant)

Polar coordinates of (-1,-1) with -\pi<\theta\le\pi?

\left(\sqrt2,-\frac{3\pi}{4}\right)

What is the curve r=a?

A circle, centre the pole, radius a.

What is the curve r=2a\cos\theta?

A circle of radius a through the pole, centre (a,0).

What is r=p\sec(\alpha-\theta)?

A straight line at perpendicular distance p from the pole.

Shape and key values of r=a(1+\cos\theta)?

Cardioid: greatest r=2a at \theta=0, r=0 at \theta=\pi.

Does r=a(3+2\cos\theta) pass through the pole?

No. r ranges from a to 5a, so it never reaches the pole.

What does r=a\cos^2\theta look like?

Two lobes along the x-axis touching at the pole (r=0 at \theta=\pm\frac{\pi}{2}).

For which \theta does r^2=a^2\cos2\theta exist?

Where \cos2\theta\ge0: |\theta|\le\frac{\pi}{4} and \frac{3\pi}{4}\le\theta\le\frac{5\pi}{4}.

Which symmetry does an equation in \cos\theta give?

Symmetry about the initial line, since \cos(-\theta)=\cos\theta.

Polar coordinates and polar curvesEdexcel A-Level Further Maths: Flashcards

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Formulae to convert polar $(r,\theta)$ to Cartesian?

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Formulae to convert polar $(r,\theta)$ to Cartesian?: $x=r\cos\theta$ , $y=r\sin\theta$
Formulae to convert Cartesian to polar?: $r=\sqrt{x^2+y^2}$ , $\tan\theta=\frac yx$ (check the quadrant)
Polar coordinates of $(-1,-1)$ with $-\pi<\theta\le\pi$ ?: $\left(\sqrt2,-\frac{3\pi}{4}\right)$
What is the curve $r=a$ ?: A circle, centre the pole, radius $a$ .
What is the curve $r=2a\cos\theta$ ?: A circle of radius $a$ through the pole, centre $(a,0)$ .
What is $r=p\sec(\alpha-\theta)$ ?: A straight line at perpendicular distance $p$ from the pole.
Cartesian equation of $r=3\sec\theta$ ?: $x=3$
What is the curve $r=k\theta$ ?: An Archimedean spiral.
Shape and key values of $r=a(1+\cos\theta)$ ?: Cardioid: greatest $r=2a$ at $\theta=0$ , $r=0$ at $\theta=\pi$ .
Does $r=a(3+2\cos\theta)$ pass through the pole?: No. $r$ ranges from $a$ to $5a$ , so it never reaches the pole.
What does $r=a\cos^2\theta$ look like?: Two lobes along the $x$ -axis touching at the pole ( $r=0$ at $\theta=\pm\frac{\pi}{2}$ ).
For which $\theta$ does $r^2=a^2\cos2\theta$ exist?: Where $\cos2\theta\ge0$ : $|\theta|\le\frac{\pi}{4}$ and $\frac{3\pi}{4}\le\theta\le\frac{5\pi}{4}$ .
Which symmetry does an equation in $\cos\theta$ give?: Symmetry about the initial line, since $\cos(-\theta)=\cos\theta$ .

Exam questions on Polar coordinates and polar curves

The point $P$ has polar coordinates $\left(4,\frac{2\pi}{3}\right)$ and the point $Q$ has Cartesian coordinates $(-1,-1)$ .
Find the polar coordinates $(r,\theta)$ of $Q$ , with $r>0$ and $-\pi<\theta\le\pi$ .2 marks
The curve $C$ has polar equation $r=4\cos\theta$ , for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$ .
The circle with polar equation $r=2$ meets $C$ at two points. Find their polar coordinates, with $r>0$ .2 marks
The curve $C$ has polar equation $r=2(1+\cos\theta)$ for $-\pi<\theta\le\pi$ .
(i) Show that $C$ is symmetrical about the initial line. (ii) State the greatest value of $r$ and the value of $\theta$ at which it occurs.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).