SOLUTION: If 1, Omega, omega^2 are cube roots of unity, prove that 1, Omega, omega^2 are vertices of an equilateral triangle.

Question 1117926: If 1, Omega, omega^2 are cube roots of unity, prove that 1, Omega, omega^2 are vertices of an equilateral triangle.
Found 2 solutions by greenestamps, math_helper:
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

What kind of "proof" are you looking for?

DeMoivre's Theorem says that the three cube roots of unity have equal moduli and are spaced at equal angles around the origin. That makes the three roots the vertices of an equilateral triangle.
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
Let the roots be where k=0,1,2.

The three roots can be found by for k=0,1,2

Using this property:

for k=0,1,2

k=0:
k=1:
k=2:

Now if you plot the real values on the x-axis and imaginary parts along the y-axis, and you connect each point to form a triangle, you will find the distance between each of the three points (vertices) is units. Thus completing the proof. There may be a more efficient method, but this method does work.

RELATED QUESTIONS
A finite set (Omega) has n elements. Show that if we count the empty set and (Omega) as... (answered by stanbon)
hello mam\sir i have some doubts in following questions 1.find the cube root of... (answered by khwang)
find the exact value of the indicated trigonometric function of omega, if omega is an... (answered by lwsshak3)
a bottle of fish oil contains 60 soft gels, each containing 800mg of the omega-3 fatty... (answered by TimothyLamb)
Find theta if omega=100 revolutions per second and t=2... (answered by Alan3354)
Could someone outline a strategy for solving the following problem? *Let \( \triangle... (answered by CPhill)
If sin O= -8/17 and tan O is positive, what is the value of cos O?... (answered by lwsshak3)
Given that radian=pi/8, use the formula omega=radian/t to find omega in... (answered by Alan3354)
If arg((z-w)/(z-w^2)) =0,then prove that re(z)=-1/2,where (w and w^2 are non real cube... (answered by richard1234)