Solving z^3 = i using De Moivre's Theorem

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This should help you find the values of R and \theta that satisfy the equation. In summary, using de Moivre's theorem and solving in polar form, the solution to the equation z^3 = i is CIS pi/6, CIS 5pi/6, and CIS (-pi/6). The modulus is 1 and the angles are pi/6, 5pi/6, and -pi/6.
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Homework Statement


solve below using de Moivre's theorem, in polar form
z^3 = i


Homework Equations


r CIS theta

Answer:CIS pi/6, CIS 5pi/6, CIS (-pi/6)

The Attempt at a Solution


r^3 = sqrt(0^2+1^2)
r^3 = sqrt (1)
r = sqrt (1)

no idea how to get the angle
 
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Mathysics said:

Homework Statement


solve below using de Moivre's theorem, in polar form
z^3 = i


Homework Equations


r CIS theta

Answer:CIS pi/6, CIS 5pi/6, CIS (-pi/6)

The Attempt at a Solution


r^3 = sqrt(0^2+1^2)
r^3 = sqrt (1)
r = sqrt (1)

no idea how to get the angle

It is not true that i = 1, which is essentially what you have done in solving the equation. 1 is simply the modulus of the complex number i.

Try writing the complex number i in polar form i.e. in the form [tex]R cis \theta[/tex], and then taking the cube root of both sides of the equation.
 

1. What is De Moivre's Theorem?

De Moivre's Theorem is a mathematical theorem that allows us to raise any complex number to a power, using trigonometric functions.

2. How is De Moivre's Theorem used to solve equations?

De Moivre's Theorem is used to solve equations involving complex numbers by converting them into trigonometric form and then using trigonometric identities to find the solutions.

3. What is the process for solving z^3 = i using De Moivre's Theorem?

The first step is to rewrite z^3 = i in trigonometric form, which is z = (cos(theta) + i*sin(theta)). Then, we can use the formula z = (cos(theta/n) + i*sin(theta/n))^n to find the three solutions. Finally, we can convert the solutions back to rectangular form if needed.

4. Can De Moivre's Theorem be used to solve other types of equations?

Yes, De Moivre's Theorem can be used to solve any equation involving powers of complex numbers, such as z^4 = -1 or z^5 = 5+5i.

5. What are the advantages of using De Moivre's Theorem to solve equations?

De Moivre's Theorem provides a systematic and efficient way to solve equations involving complex numbers. It also allows us to find multiple solutions to an equation, rather than just one.

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