Solving ( z cdot i^3 1 ) Using De Moivre's Theorem

In this article, we will explore how to solvethe equation ( z cdot i^3 1 ) using De Moivre's Theorem. De Moivre's Theorem is a powerful tool in complex analysis that relates the trigonometric form of complex numbers to their exponentiation. This theorem is particularly useful for finding roots of complex numbers. We will delve into the details and provide step-by-step solutions to gain a comprehensive understanding of the process.

Introduction to De Moivre's Theorem

De Moivre's Theorem states that for any real number x and integer n, the following holds true:

e^{ix} cos(x) isin(x)

This theorem allows us to express complex numbers in polar form and manipulate them efficiently. By understanding this theorem, we can solve various problems involving complex numbers and trigonometric identities.

Solving ( z cdot i^3 1 )

To solve the equation z cdot i^3 1, we first need to understand the properties of i. We know that:

i 1 quad text{arg} , i frac{pi}{2}

Given the equation, we can express the right-hand side in exponential form:

1 e^{i cdot 0}

Using De Moivre's Theorem, we can write:

z cdot i^3 z cdot e^{i cdot frac{3pi}{2}k}end{code>

where k is an integer. We can rewrite the original equation as:

z cdot e^{i cdot frac{3pi}{2}k} e^{i cdot 0}

From this equation, we can express z as:

z e^{-i cdot frac{3pi}{2}k} cdot 1end{code>

For simplicity, let's focus on k 0, 1, 2 to find the distinct cube roots of i:

z e^{-i cdot frac{3pi}{2}k} end{code>

Let's evaluate this for each value of k.

Finding the Cube Roots of ( i )

For k 0:

z_0 e^{-i cdot frac{3pi}{2} cdot 0} cos(0) isin(0) 1

For k 1:

z_1 e^{-i cdot frac{3pi}{2} cdot 1} cos(-frac{3pi}{2}) isin(-frac{3pi}{2}) 0 iend{code>

For k 2:

z_2 e^{-i cdot frac{3pi}{2} cdot 2} cos(-3pi) isin(-3pi) -1

These are the distinct cube roots of ( i ).

Deriving the Solutions Explicitly

Given the roots, we can find the explicit form of z in Cartesian form:

z_0 1end{code>

z_1 0 i iend{code>

z_2 -1end{code>

Hence, the solution set is:

{1, i, -1}end{code>

Conclusion

In this article, we explored how to solve the equation z cdot i^3 1 using De Moivre's Theorem. We derived the distinct roots of ( i ) and provided explicit forms. De Moivre's Theorem is a powerful tool in complex analysis, and its applications are widespread in various fields, including engineering, physics, and mathematics.