CAIE A-Level Further Math AS 1.5.1 Polar Coordinates Questions

Practise converting between Cartesian and polar equations and interpreting polar curves.

Exam points

convert between Cartesian and polar equations
find polar coordinates and curve properties
analyse polar curves in context

[Maximum number: 4]

Show that the curve with Cartesian equation

\left(x^{2}+y^{2}\right)^{\frac{5}{2}}=4 x y\left(x^{2}-y^{2}\right)

has polar equation $r=\sin 4 \theta$ .

The curve C has polar equation $r=\sin 4 \theta$ , for $0 \leqslant \theta \leqslant \frac{1}{4} \pi$ .

$r^{5}=4 x y\left(x^{2}-y^{2}\right)$
Uses $r^{2}=x^{2}+y^{2}$ .

$r^{5}=4 r^{4} \cos \theta \sin \theta\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$
Uses $x=r \cos \theta$ and $y=r \sin \theta$ .

$r=2 \sin 2 \theta \cos 2 \theta$
Applies at least one double angle formula.

$r=\sin 4 \theta$
Applies both double angle formulae, AG.

4

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CAIE A-Level Further Math AS 1.5.1 Polar Coordinates Questions question 1