# De Moivre's formula

There are two distinct complex numbers z such that z3 is equal to 1 and z is not equal 1.

Calculate the sum of these two numbers.

Correct result:

S =  -1

#### Solution:

$z^3 = 1 \ \\ 1 = 1(\cos 0 + i \sin 0 ) \ \\ z_k = \sqrt{1}( \cos( \dfrac{2\pi k }{3} ) + i \sin( \dfrac{2\pi k }{3} ) ) \ \\ z_1 = 1 \ \\ z_2 = \cos( 120 ^\circ ) + i \sin( 120 ^\circ ) = -0.5 + 0.86602540378444 i \ \\ z_3 = \cos( 240 ^\circ ) + i \sin( 240 ^\circ ) = -0.5 - 0.86602540378444 i \ \\ S = z_2 + z_3 = -1$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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