Polar coordinate identities

There’s a few things that we can say are true when working with polar coordinates - identities.

They’re really useful especially later on, when we’ll use them to convert from polar forms to cartesian forms, and the other way round.

x=r\cos\theta

We know this because \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}, and in this case the adjacent side is x and the hypotenuse is r.

So \cos\theta=\frac{x}{r}, and rearranging gives us x=r\cos\theta.

y=r\sin\theta

We know this one because \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}, and in this case the opposite side is y and the hypotenuse is r. So \sin\theta=\frac{y}{r}, and rearranging gives us y=r\sin\theta.

r^2=x^2+y^2

• We can rewrite this one to r=\sqrt{x^2+y^2} if absolutely required.

\tan\theta=\frac yx

• We can rearrange this as either of:
  • y=x\tan\theta
  • x=\frac y{\tan\theta}
  • \theta=\tan^{-1}\left(\frac yx\right)=\arctan\left(\frac yx\right)

We do also need to check which quadrant it is in, to decide on what becomes negative in the polar coordinates.

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