Convert Cartesian to polar.

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

What is the curve r=a?

A circle, centre the pole, radius a.

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

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

What is the curve \theta=\alpha?

A half-line from the pole at angle \alpha to the initial line.

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

A straight line at perpendicular distance p from the pole; Cartesian form x\cos\alpha+y\sin\alpha=p.

Name the curve r=a(1+\cos\theta).

A cardioid, with a cusp at the pole when \theta=\pi.

Why does r=a(3+2\cos\theta) have no cusp?

r\ge a0 for all \theta, so it never reaches the pole.

What shape is r^2=a^2\cos2\theta and where does it exist?

A lemniscate (figure of eight), where \cos2\theta\ge0: -\frac{\pi}{4}\le\theta\le\frac{\pi}{4} and \frac{3\pi}{4}\le\theta\le\frac{5\pi}{4}.

Condition for a tangent parallel to the initial line?

\frac{d}{d\theta}(r\sin\theta)=0.

Condition for a tangent perpendicular to the initial line?

\frac{d}{d\theta}(r\cos\theta)=0.

Polar coordinates and curve sketchingEdexcel International A Level Further Maths: Flashcards

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Convert polar $(r,\theta)$ to Cartesian.

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Convert polar $(r,\theta)$ to Cartesian.: $x=r\cos\theta$ , $y=r\sin\theta$ .
Convert Cartesian to polar.: $r^2=x^2+y^2$ , $\tan\theta=\frac yx$ (check the quadrant).
What is the curve $r=a$ ?: A circle, centre the pole, radius $a$ .
What is the curve $r=2a\cos\theta$ ?: A circle, radius $a$ , centre $(a,0)$ , through the pole.
What is the curve $\theta=\alpha$ ?: A half-line from the pole at angle $\alpha$ to the initial line.
What is $r=p\sec(\alpha-\theta)$ ?: A straight line at perpendicular distance $p$ from the pole; Cartesian form $x\cos\alpha+y\sin\alpha=p$ .
What is the curve $r=k\theta$ ?: An Archimedean spiral.
Name the curve $r=a(1+\cos\theta)$ .: A cardioid, with a cusp at the pole when $\theta=\pi$ .
Why does $r=a(3+2\cos\theta)$ have no cusp?: $r\ge a>0$ for all $\theta$ , so it never reaches the pole.
What shape is $r=a\cos2\theta$ ?: A four-petalled rose.
What shape is $r^2=a^2\cos2\theta$ and where does it exist?: A lemniscate (figure of eight), where $\cos2\theta\ge0$ : $-\frac{\pi}{4}\le\theta\le\frac{\pi}{4}$ and $\frac{3\pi}{4}\le\theta\le\frac{5\pi}{4}$ .
Condition for a tangent parallel to the initial line?: $\frac{d}{d\theta}(r\sin\theta)=0$ .
Condition for a tangent perpendicular to the initial line?: $\frac{d}{d\theta}(r\cos\theta)=0$ .

Exam questions on Polar coordinates and curve sketching

The straight line $l$ has polar equation $r=4\sec\left(\theta-\frac{\pi}{3}\right)$ , where $r\ge0$ and the pole is $O$ .
Find the polar coordinates of the point where $l$ meets the initial line.2 marks
The curve $S$ has polar equation $r=3\theta$ , for $\theta\ge0$ .
Find the Cartesian coordinates of the point on $S$ where $\theta=\frac{3\pi}{2}$ .2 marks
The curve $C$ has polar equation $r=3+2\cos\theta$ , for $0\le\theta<2\pi$ .
Find the greatest and least values of $r$ , stating the value of $\theta$ at each, and explain why $C$ does not pass through the pole.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).