Defining Polar Coordinates (College Board AP® Calculus BC): Revision Note

Defining polar coordinates

What are polar coordinates?

• Polar coordinates are an alternative way to describe the position of points and curves in two-dimensions using

  • a distance, $r$

    • where $r$ can be positive or negative

    • measured from an origin, called a pole

  • and an angle, $\theta$

    • measured in radians from an initial line (extending from the pole)

    • The initial line is usually the $x$-axis

• The polar coordinates of a point are written as $\left(r,\theta \right)$

  • These are not to be confused with a point given in $\left(x,y\right)$ form

• Negative values of $r$ are drawn in the opposite direction to $\theta$

  • $\left(-r,\theta \right)=\left(r,\theta +\pi \right)$

How are polar coordinates related to x and y coordinates?

• If a point P has polar coordinates $\left(r,\theta \right)$, then the $x$ and $y$ coordinates in the $xy$-plane are found by trigonometry:

  • $x=r\cos \theta$

  • $y=r\sin \theta$

• It can also be helpful to make $r^2$ and $\theta$ the subject to give two more relationships:

  • $r^2=x^2+y^2$

  • $\tan \theta =\frac{y}{x}$

• These are found using trigonometric identities:

  • $x^2+y^2={\left(r\cos \theta \right)}^2+{\left(r\sin \theta \right)}^2=r^2\left({\cos }^2\theta +{\sin }^2\theta \right)=r^2$

  • and $\frac{y}{x}=\frac{r\sin \theta }{r\cos \theta }=\tan \theta$

How do I sketch a polar curve?

• A polar curve is a curve given in the form $r=f\left(\theta \right)$

  • They are sometimes given in the form $r^2=g\left(\theta \right)$

• The polar curve may be restricted to a range of $\theta$ values, e.g. $0\le \theta \le \frac{\pi }{2}$

• To sketch a polar curve, use the graph sketching function on your calculator

  • Change the type of graph to 'polar'

• An alternative way to sketch is by plotting points

  • Find the distance $r$ for some key angles, $\theta =0,\frac{\pi }{6},\frac{\pi }{3},...2\pi$

    • Plot these points on a polar grid (see below)

• The following are some examples of polar curves (you do not need to learn these):

• Note that the spiral $r=k\theta$ has an angle $\theta$ that can continue beyond $2\pi$

How do I find x and y coordinates on a polar curve?

• To find the $x$ and $y$ coordinates of a point on a polar curve, substitute the equation of the polar curve, $r=f\left(\theta \right)$, into the trigonometric relationships $x=r\cos \theta$ and $y=r\sin \theta$ to get:

  • $x=f\left(\theta \right)\cos \theta$

  • $y=f\left(\theta \right)\sin \theta$

• Different values of $\theta$ will give different $x$ and $y$ coordinates

How do I find the average distance from the origin to a point on a polar curve?

• The average distance from the origin to a point on a polar curve, $r\left(\theta \right)$, where $\alpha \le \theta \le \beta$ is:

$\frac{1}{\beta -\alpha }{\int }_{\alpha }^{\beta }r\left(\theta \right)d\theta$

How do I convert a polar equation into (and out of) x and y coordinates?

• To convert an equation from polar coordinates into $x$ and $y$ coordinates, use the algebraic relationships above

  • e.g. $r=4\cos \theta$

    • Multiply both sides by $r$ to get $r^2=4r\cos \theta$

    • Use that $r^2=x^2+y^2$ and that $r\cos \theta =x$ to give $x^2+y^2=4x$

    • By completing the square, this is ${\left(x-2\right)}^2+y^2=2^2$

    • which is a circle, radius 2, centre $\left(2,0\right)$

  • Note that $r^2=x^2+y^2$ means $r=\pm \sqrt{x^2+y^2}$

    • Only use the negative square root for parts of the curve where $r$ goes negative

• To convert an equation from $x$ and $y$ into polar coordinates, simply substitute in $x=r\cos \theta$ and $y=r\sin \theta$ and rearrange

  • You need to be able to recognize the following key examples: