9231. Further Pure 1. Polar Coordinates

Instead of measuring a point on a plane as two distances, (x,y), it can be represented by:
- a distance, r (from the origin, now called a pole); and
- an angle 𝜽 (measured anticlockwise from initial line OX).

What is x in terms of r and 𝜽 ?

What is y in terms of r and 𝜽 ?

And therefore:

What is r in terms of x and y?

What is 𝜽 in terms of x and y? (But always check that the 𝜽 you get is in the right quadrant. If not, you will need to add or subtract π)

We write polar equations in the form r=f(𝜽)

So:
- x=f(𝜽)cos𝜽; and
- y=f(𝜽)sin𝜽.

Worked Examples

Worked Solutions

Some Basic Curves

r=k is a circle.

r=𝜽 is an outward spiral starting from the pole.

For more complex functions we need to draw a table of values for different 𝜽’s to understand how the curve behaves (but we can also consider properties such as symmetries and behaviours of known functions)

Sketching

We only sketch parts of the curve where r is greater than or equal to zero. We can choose the domain of 𝜽 to ensure this, e.g. r=cos(𝜽) should have domain [-π/2 , π/2];
Increasing 𝜽 on a curve effectively rotates the curve.
Polar curves containing factors of cos(𝜽) will have a line of symmetry at 𝜽=0. (because cos(-𝜽)=cos(𝜽))

Worked Example

Worked Solution

Worked Example

N.B. By convention, any points for which r<0 are not plotted

Worked Solution

Exercise 1 (can check curves on Geogebra)

Answers to Exercise 1

Worked Solutions to Exercise 1

Intersection of Curves

Polar curves intersect as do rectangular ones, i.e. r=f(𝜽) intersects with r=g(𝜽) when f(𝜽)=g(𝜽)

We can solve this equation to find (r,𝜽) at the point of intersection

Always sketch the given curves to see how they interact in the polar plane.

Area under Curve

$A = \int_{\alpha}^{\beta}{\frac{1}{2}r^2 d \theta}$

When applying this formula, it is important to remember that r must be defined and non-negative throughout the interval 𝝰≤𝜽≤β.

Additional Notes

Symmetries in a curve can be used to take limits over a reduced area and then multiply up by the amount of times this area repeats.

As with integration of rectangular curves, the area between curves is the difference between the area of the curves. If the limits differ for two curves, the integrals may not be combined and must be calculated separately.

At the maximum distance from the pole, $\frac{dr}{d \theta } = 0$ . This can be helpful for finding the tips of loops.

We can also calculate the maximum and minimum x-values and y-values of parametric curves, by calculating $\frac{dx}{d \theta }$ and $\frac{dy}{d \theta }$ (using x=f(𝜽)cos𝜽 and y=f(𝜽)sin𝜽)