- 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 π½ (measured anticlockwise from
**initial line**OX).

- a distance, r (from the origin, now called a

What is x in terms of r and π½ ?

What is y in terms of r and π½ ?

And therefore:

What is r in terms of x and y?

What is π½ in terms of x and y? (But always check that the π½ you get is in the right quadrant. If not, you will need to add or subtract Ο)

We write polar equations in the form r=f(π½)

- So:
- x=f(π½)cosπ½; and
- y=f(π½)sinπ½.

**Worked Examples**

**Worked Solutions**

**Some Basic Curves**

r=k is a circle.

r=π½ is an outward spiral starting from the pole.

For more complex functions we need to draw a table of values for different π½βs 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 π½ to ensure this, e.g. r=cos(π½) should have domain [-Ο/2 , Ο/2];
- Increasing π½ on a curve effectively rotates the curve.
- Polar curves containing factors of cos(π½) will have a line of symmetry at π½=0. (because cos(-π½)=cos(π½))

**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(π½) intersects with r=g(π½) when f(π½)=g(π½)

We can solve this equation to find (r,π½) at the point of intersection

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

**Area under Curve**

When applying this formula, it is important to remember that r must be defined and non-negative throughout the interval π°β€π½β€Ξ².

**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, . 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 and (using x=f(π½)cosπ½ and y=f(π½)sinπ½)

**Worked Example**

Find the total area of the two loops of the curve r = acos2π½, where a>0 and -Ο < π½ < Ο:

**Worked Solution**

**Exercise 2 and General Exercises**

**Answers to Exercise 2 and General Exercise**

Worked Solutions to Exercise 2

**Relevant Past Paper Questions**

Summer 2020 Paper 11:

Summer 2020 Paper 11 Mark Scheme

Summer 2020 Paper 11:

Summer 2020 Paper 13 Mark Scheme

Winter 2020 Paper 11:

Winter 2020 Paper 11 Mark Scheme

Winter 2020 Paper 12: