For each value of n, how is the length of the side of a polygon related to the central angle subtended by the side? (The side is a chord of the unit circle.)
Hello,
I had an assignment that required me to solve for the roots of unity for various equations of the form z^n-1=0. Then , I was asked to represent the roots of unity for each equation on an argand diagram in the form of a regular polygon.
I did all of that , but I have a critical question ,
is there a relation ship between the power n and the length of the polygon side?
I mean , for the euqation Z^3 - 1 = 0 for example, is there a formula relating the power to the lenght of one side of the polygon ?
Thanks.
1 can be represented in polar form in the obvious way: . The nth roots are given by for k= 0 to n- 1. That is, 1 has n nth roots, equally spaced around the unit circle. If you draw lines from each root to the origin, you have n isosceles triangles and the angle at the vertex (the origin) is . If you draw a perpendicular from the origin to one side, it divides that isosceles triangle into two right triangles having hypotenuse 1 and angle . The "opposite side" to that angle is given by [itex]sin(\frac{\pi}{n})[/tex] and is 1/2 the length of one side of the polygon. The length of one side, then, is .
As a check note that if n= 4, the polygon is a square, having diagonals of length 2. while, by the Pythagorean theorem so , .
Similarly, if n= 6, the polygon is a hexagon having diagonals of length 2. .
Thanks for your generous contribution sir ,
In the same context , I was asked to obtain solutions for the equation Z^n-i = 0 for three cases where n = 1,2,3
I did that for all of them , for example when n = 3 the solutions set was :
i^(1/3) = 1^(1/3) e^[i(π/2 + 2kπ)]/3 = 1 e^[i(π/6 + 2kπ/3)] , where k = 0, 1, and 2 ... and I followed the same approach for n = 4 and 5.
Then I had to plot each of those roots on an argand diagram , and I did that as well.
After that , I was asked to generalize my and prove my results for Z^n= a+bi , where | a+bi | =1 . how can I do that , shall I follow the above method as well or there is a different generalization? Thanks
Yes , sir..thanks but I would like to calrify the question more :
First , it is required to generalize and prove the resuls for z when z^n = cos (x) + isin (X)
Secondly . it is required to generalize for Z when z^n = cos (Kx) + isin (Kx)
Thanks and sorry for disturbance.
Sorry I made a mistake ...
First thing reqired is to generalize and prove the results for z when z^n = cis(x) (or | a+bi | =1)
Second thing is what hepppens when Z^n = cis(x0 (or | a+bi | does not equal 1 )
Sorry again.