Polar Coordinates
The Cartesian system in two-dimensions models points in terms of x and y. The polar system, however, models a point as a distance form the pole, r (generally the origin) at a certain angle from the initial line, θ (typically the positive horizontal axis). Yes, this is like the modulus-argument form of complex numbers and Argand diagrams.
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_5921f291285645409bb8968cdd5a994d~mv2.jpg/v1/fill/w_112,h_105,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_5921f291285645409bb8968cdd5a994d~mv2.jpg)
From the diagram, we can derive equations to convert between polar and Cartesian systems:
r cos(θ) = x
r sin(θ) = y
Where θ is given by:
θ = arctan(y/x)
And r is defined using Pythagoras' theorem:
r² = x² + y²
Sketching Polar Curves
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_fbcd38acccda4d788b4cd0ec2683efe7~mv2.jpg/v1/fill/w_147,h_57,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_fbcd38acccda4d788b4cd0ec2683efe7~mv2.jpg)
To sketch a polar curve, use a graphical calculator or draw u a table of values for regular intervals of θ. This can be done quickly using the table function on the CASIO ClassWiz fx-991EX, and we recommend using π/6 as an interval.
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_46fb4492d19c42c6b736a0b7d77cf9d0~mv2.jpg/v1/fill/w_139,h_174,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_46fb4492d19c42c6b736a0b7d77cf9d0~mv2.jpg)
The curve in this example is known as a cardioid, due to its dimple. This is common for equations in the form r = a(p+qcos(θ)), but only if q ≤ p < 2q. When p ≥ 2q, there is no dimple, making it more egg-shaped:
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_6d03654d142c4249a9f2fb8b42b68fdb~mv2.jpg/v1/fill/w_126,h_82,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_6d03654d142c4249a9f2fb8b42b68fdb~mv2.jpg)
Areas Enclosed by Polar Curves
The area of a sector of a polar curve can be calculated using integration. However, simply integrating r will not work. Instead:
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_58b25182e9a044c7a4a8259a2902dcc7~mv2.jpg/v1/fill/w_105,h_56,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_58b25182e9a044c7a4a8259a2902dcc7~mv2.jpg)
Of course, you an also calculate areas between polar curves. To do this, you need to find the angle at which they intersect.
Tangents to Polar Curves
To find tangents to a polar curve, you need to convert it into Cartesian form (one equation for x and one for y, both in terms of θ), using the formula at the top of this notes sheet. Then, you can differentiate parametrically.
Standard results are:
![Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_0c2315fa8cac4dee8ed4e7fc81cbe540~mv2.jpg/v1/fill/w_147,h_39,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_0c2315fa8cac4dee8ed4e7fc81cbe540~mv2.jpg)