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