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Hey, cheers for the reply. I've drawn out the rough movement of the ant, but I don't really know what to do with this information.
It travels the first leg of the journey to the point (0,1).
It then turns 20 degrees to the right, and travels another one unit.
So let's consider this shortened jouney. Are you able to tell me what the position vector of the ant is after these two legs?
The main idea hinges on the fact that if you travel (a,b) in one direction, then you travel by a vector (c,d) in another direction, then your final position is (a+c,b+d). You should be able to tell me the final position of the ant after these two legs. You should also employ trigonometry to determine the horizontal/vertical components of the moment that happen on the second leg of the journey.
Next you have (z^4-1)/(z-1) where z = e^(i*2pi/9) this can be rearranged to: [[z^2 - 1/(z^2)]/[z^0.5-z^0.5]]*(z^(3/2). Using de moivres this is: [sin(4pi/9)/sin(pi/9)]*e^(i*pi/3). This is the addition of the complex numbers and is the complex number from the origin of the argand diagram. Clearly, the distance from the origin is the modulus of the complex number which is: [sin(4pi/9)/sin(pi/9)]