# Maths helppp

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#1
I can't seem to understand what this question wants from me, so if someone could please point me in the right direction, it would really help!!
It's number 6 btw.
https://imageshack.com/a/img922/8481/Es1lSX.jpg

Thanks!
Last edited by nabsers; 3 years ago
0
3 years ago
#2
(Original post by nabsers)
I can't seem to understand what this question wants from me, so if someone could please point me in the right direction, it would really help!!
It's number 6 btw.
https://imageshack.com/a/img922/8481/Es1lSX.jpg

Thanks!
It would be useful to draw a small diagram representing the journey of the ant over these four legs.
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#3
(Original post by RDKGames)
It would be useful to draw a small diagram representing the journey of the ant over these four legs.
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.
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3 years ago
#4
(Original post by nabsers)
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.
Suppose the ant starts at (0,0).

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.
0
3 years ago
#5
I remeber the book wants you to use this weird way with complex numbers and none of our class really got it, but then we tried just drawing out the path and used just triangles and trig etc and it gave you the answer so much easier
0
3 years ago
#6
If you plot it on an argand diagram it becomes simpler. The and moves unit distance one from the argand diagram. Then it moves along the complex number which has an argument of 2pi/9 and modulus 1. Then it moves along a complex number which has argument 4pi/9 and modulus 1. Then it moves along a complex number argument 6pi/9. Hence, you have the following addition of complex numbers: 1 + e^(-2pi/9) + e^(-4pi/9) + e^(-6pi/9). For simplicities sake, reflect the whole diagram through the real line. Hence, the addition series becomes: 1 + e^(2pi/9) + e^(4pi/9) + e^(6pi/9). Then take the summation of the series.

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)]
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