A-level Mathematics/MEI/FP2/Polar Co-ordinates

Cartesian coordinates are a system of looking up a point on a grid relative to the origin, at 0,0. You can express Cartesian coordinates in the format (x,y).

Polar coordinates are another system, but rather in terms of perpendicular distance, you use the Pythagorean distance and the angle. You can express Polar coordinates in the format (r, θ) where r is the radius, or distance, and θ (theta) is the angle. Angles are generally expressed in radians anticlockwise from the x axis. For instance, the point (1,1) in Cartesian coordinates, would translate to (√2,π/4) in polar coordinates.

To translate set of Cartesian coordinates into Polar coordinates:

$$r= \sqrt{x^2 + y^2}$$

$$\tan \theta= (y/x)$$

Where x=0, θ=0, but if x=0 and y=0, θ is undefined.

Rearranging $$\tan \theta= (y/x)$$ for θ yields $$\theta = \arctan(y/x)$$, however this will always give two values as a $$\arctan(positive)$$ could mean that x=negative and y=negative or that x=positive and y=positive, and $$\arctan(negative)$$ could mean that x=positive and y=negative or x=negative and y=positive. So you need to check the quadrants.

Likewise, to convert a set of Polar coordinates into Cartesian:

$$x= r \cos \theta $$

$$y= r \sin \theta $$