This is a question on the application of complex numbers.

The standard solution requires us to formulate a function to represent the action of moving forward one unit and turning to the right by 2π/9, and compute the sum using the sum of geometric series formula.

But there’s a more elegant way to do this, which lets us avoid all that complicated looking formula.

Can you figure this out?

Here’s a hint: try using drawing a nonagon!

When Geometry Beats Complex Numbers

After pondering for a few minutes, I decided to draw the diagram and found out it was a regular nonagon or a 9-gon. Angle chasing gives us the values of the required angles. We then use the fact that the blue line has the same length as the green line. Finally, the sine rule allows us to arrive at our solution.

Did you figure it out?