Cartesian to polar conversion?

r=\sqrt{x^2+y^2}, \tan\theta=\frac yx with the quadrant checked.

Why must you check the quadrant when finding \theta?

\tan\theta=\frac yx has two solutions per turn, so the calculator can return the wrong one.

What is the initial line?

The positive x-axis, from which \theta is measured.

Convert (-\sqrt3,-1) to polar form with -\pi<\theta\le\pi.

\left(2,-\frac{5\pi}{6}\right).

How do you convert r=6\cos\theta to Cartesian form?

Multiply by r: r^2=6r\cos\theta, so x^2+y^2=6x.

Which identity is used to convert r^2=18\sin2\theta?

\sin2\theta=2\sin\theta\cos\theta, then multiply by r^2 to get (x^2+y^2)^2=36xy.

Polar and Cartesian coordinatesAQA A-Level Further Maths: Flashcards

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What do $r$ and $\theta$ mean in polar coordinates?

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What do $r$ and $\theta$ mean in polar coordinates?: $r$ is the distance from the pole; $\theta$ is the anticlockwise angle from the initial line.
Polar to Cartesian conversion?: $x=r\cos\theta$ , $y=r\sin\theta$ .
Cartesian to polar conversion?: $r=\sqrt{x^2+y^2}$ , $\tan\theta=\frac yx$ with the quadrant checked.
Why must you check the quadrant when finding $\theta$ ?: $\tan\theta=\frac yx$ has two solutions per turn, so the calculator can return the wrong one.
What is the pole?: The origin, the point $r=0$ .
What is the initial line?: The positive $x$ -axis, from which $\theta$ is measured.
Polar form of $x^2+y^2$ ?: $r^2$ .
Polar equation of the circle $x^2+y^2=4x$ ?: $r=4\cos\theta$ .
Convert $(-\sqrt3,-1)$ to polar form with $-\pi<\theta\le\pi$ .: $\left(2,-\frac{5\pi}{6}\right)$ .
Cartesian coordinates of $\left(4,\frac{5\pi}{6}\right)$ ?: $(-2\sqrt3,\,2)$ .
How do you convert $r=6\cos\theta$ to Cartesian form?: Multiply by $r$ : $r^2=6r\cos\theta$ , so $x^2+y^2=6x$ .
Cartesian form of $r(\cos\theta+\sin\theta)=6$ ?: $x+y=6$ .
Which identity is used to convert $r^2=18\sin2\theta$ ?: $\sin2\theta=2\sin\theta\cos\theta$ , then multiply by $r^2$ to get $(x^2+y^2)^2=36xy$ .

Exam questions on Polar and Cartesian coordinates

The point $P$ has polar coordinates $\left(4,\dfrac{5\pi}{6}\right)$ .
The point $P'$ is the reflection of $P$ in the initial line. Give the polar coordinates of $P'$ with $r>0$ and $-\pi<\theta\le\pi$ .2 marks
The point $R$ has Cartesian coordinates $\left(-\sqrt3,-1\right)$ .
The point $R$ is rotated through $\frac{\pi}{2}$ anticlockwise about the origin. Find the Cartesian coordinates of its new position.2 marks
Two curves are given. Curve $C$ has Cartesian equation $x^2+y^2=4x$ and curve $D$ has polar equation $r^2=18\sin2\theta$ .
Show that $C$ has polar equation $r=4\cos\theta$ .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).