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:

z3=1 1=1(cos⁡0+isin⁡0) zk=13(cos⁡(2πk3)+isin⁡(2πk3)) z1=1 z2=cos⁡(120∘)+isin⁡(120∘)=−0.5+0.86602540378444i z3=cos⁡(240∘)+isin⁡(240∘)=−0.5−0.86602540378444i S=z2+z3=−1z^3 = 1 \ \\ 1 = 1(\cos 0 + i \sin 0 ) \ \\ z_k = \sqrt[3]{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

Try another example


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!


Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




Tips to related online calculators
Try our complex numbers calculator.

You need to know the following knowledge to solve this word math problem:

Next similar math problems:

  • Difference AP
    delta Calculate the difference of arithmetic progression if the sum of its first 19 members Sn = 8075 and the first member is a1 = 20
  • Linear imaginary equation
    cplx_function Given that ? "this is z star" Find the value of the complex number z.
  • Complex number coordinates
    complex_numbers Which coordinates show the location of -2+3i
  • Imaginary numbers
    Complex_conjugate_picture.svg Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum?
  • Reciprocal
    complex_numbers Calculate reciprocal of z=0.8-1.8i:
  • Is complex
    cplx Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex?
  • Im>0?
    ReIm Is -10i a positive number?
  • Log
    exp_log Calculate value of expression log |3 +7i +5i2| .
  • Let z1=x1+y1i
    cplx_function Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)
  • Median and modus
    dice_3 Radka made 50 throws with a dice. The table saw fit individual dice's wall frequency: Wall Number: 1 2 3 4 5 6 frequency: 8 7 5 11 6 13 Calculate the modus and median of the wall numbers that Radka fell.
  • Answer
    juice Mam used 9 red, 9 green apples and 3 pears to prepare juice. How many pieces of fruit used by mom to prepare this juice?
  • Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  • 75th percentile (quartille Q3)
    statistics Find 75th percentile for 30,42,42,46,46,46,50,50,54
  • Vectors
    green For vector w is true: w = 2u-5v. Determine coordinates of vector w if u=(3, -1), v=(12, -10)
  • Combine / add term
    combine Combine like terms 4c+c-7c
  • Discriminant
    Quadratic_equation_discriminant Determine the discriminant of the equation: ?
  • Calculation
    pocty How much is sum of square root of six and the square root of 225?