Introduction to Polar Coordinates
Key Questions

The rectangular coordinates
#(x,y)=(4,0)# is equivalent to the polar coordinates#(r,theta)=(4,pi)# Let us look at some details.
Since
#r# is the distance of#(4,0)# from the origin,#r=sqrt{(4)^2+0^2}=4# Since the segment from the origin to
#(4,0)# makes#180^circ=pi# rad with the positive xaxis,#theta=pi# 
Answer:
Useful applications in physics and engineering.
Explanation:
From a physicist's point of view, polar coordinates
#(r and theta)# are useful in calculating the equations of motion from a lot of mechanical systems.Quite often you have objects moving in circles and their dynamics can be determined using techniques called the Lagrangian and the Hamiltonian of a system. Using polar coordinates in favor of Cartesian coordinates will simplify things very well.
Hence, your derived equations will be neat and comprehensible .
Besides mechanical systems, you can employ polar coordinates and extend it into a 3D ( spherical coordinates ). This will help a lot in doing calculations on fields . Example: electric fields and magnetic fields and temperature fields.
In short, polar coordinates make calculation easier for physicists and engineers. Thanks to that, we have better machines and better understanding on electricity and magnetism (essential for generating power).
PS: Knowing the why's and the how's in school is important even if you are not going to use them in real life. The point is that we have to set ignorance aside and appreciate the things we take for granted. Life as we know it will never be the same without math, science and even literature. Kudos for asking this question!

Please see the video for Polar Coordinate basics.