What does \theta=\alpha look like?

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

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

A circle through the pole, diameter a along the initial line, centre \left(\frac a2,0\right).

What does r=a\sin\theta look like?

A circle through the pole, diameter a along \theta=\frac{\pi}{2}, centre \left(0,\frac a2\right).

What does r=a\theta look like?

An Archimedean spiral winding outward from the pole.

What is the shape of r=a(1+\cos\theta)?

A cardioid, symmetrical about the initial line, with a cusp at the pole.

When does r=a+b\cos\theta have an inner loop?

When a<b, because r becomes negative for some \theta.

How many petals does r=a\cos n\theta have?

n petals if n is odd; 2n petals if n is even.

What does a negative r mean when plotting?

The point is |r| from the pole in the opposite direction, at angle \theta+\pi.

How do you find where a polar curve passes through the pole?

Solve r=0 for \theta; the curve is tangent to those lines at the pole.

Which symmetry does f(-\theta)=f(\theta) show?

Symmetry about the initial line.

Which symmetry does f(\pi-\theta)=f(\theta) show?

Symmetry about the line \theta=\frac{\pi}{2}.

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

A lemniscate (figure of eight), only where \cos2\theta\ge0.

Sketching polar curvesAQA A-Level Further Maths: Flashcards

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What does $r=a$ look like?

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What does $r=a$ look like?: A circle with centre at the pole and radius $a$ .
What does $\theta=\alpha$ look like?: A half-line from the pole at angle $\alpha$ to the initial line.
What does $r=a\cos\theta$ look like?: A circle through the pole, diameter $a$ along the initial line, centre $\left(\frac a2,0\right)$ .
What does $r=a\sin\theta$ look like?: A circle through the pole, diameter $a$ along $\theta=\frac{\pi}{2}$ , centre $\left(0,\frac a2\right)$ .
What does $r=a\theta$ look like?: An Archimedean spiral winding outward from the pole.
What is the shape of $r=a(1+\cos\theta)$ ?: A cardioid, symmetrical about the initial line, with a cusp at the pole.
When does $r=a+b\cos\theta$ have an inner loop?: When $a<b$ , because $r$ becomes negative for some $\theta$ .
How many petals does $r=a\cos n\theta$ have?: $n$ petals if $n$ is odd; $2n$ petals if $n$ is even.
What does a negative $r$ mean when plotting?: The point is $|r|$ from the pole in the opposite direction, at angle $\theta+\pi$ .
How do you find where a polar curve passes through the pole?: Solve $r=0$ for $\theta$ ; the curve is tangent to those lines at the pole.
Which symmetry does $f(-\theta)=f(\theta)$ show?: Symmetry about the initial line.
Which symmetry does $f(\pi-\theta)=f(\theta)$ show?: Symmetry about the line $\theta=\frac{\pi}{2}$ .
What shape is $r^2=a^2\cos2\theta$ , and where does it exist?: A lemniscate (figure of eight), only where $\cos2\theta\ge0$ .

Exam questions on Sketching polar curves

A curve $C$ has polar equation $r=4(1+\cos\theta)$ for $0\le\theta<2\pi$ .
Explain why $C$ is symmetrical about the initial line, and find the value of $r$ when $\theta=\frac{\pi}{2}$ .2 marks
A curve $C$ has polar equation $r=3\cos2\theta$ for $0\le\theta<2\pi$ .
When $\theta=\frac{\pi}{2}$ , $r$ is negative. Find the Cartesian coordinates of the point on $C$ with $\theta=\frac{\pi}{2}$ .2 marks
A curve $C$ has polar equation $r=2+3\cos\theta$ for $0\le\theta<2\pi$ .
Find the values of $\theta$ at which $C$ passes 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).