For example, suppose we want to find a mapping from

to

. I will state this right now, there are many many such mappings, i.e. what we will find it not unique. However, we can find this mapping without any algebra! Just using the geometric facts above. The first step is to turn this line

to the line

. The reason being is because we know how to deal with the line

. (The approach I use is not unique, if you see your own way of doing it, it will also work). First, map

to

, we do this easily by

. Second, we rotate the y-axis to the x-axis, we can do this by

. Now we are at

. Third, use the function

to get the unit circle. Thus,

will do the job.