Problem 74 - Algebra and Trigonometry

Question

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex cube roots of \(i\)

Step-by-Step Solution

Answer

The complex cube roots of \(i\) are \(i/2+\sqrt{3}/2\), \(-i/2+\sqrt{3}/2\), and \(-i\).

Step 1: Express Complex Number in Polar Form

First, the given complex number \(i\) is written in its polar form or cis form which is \(cis(\theta)\). For the number \(i\), we can write it as \(cis(\pi/2)\) or \(cis(\theta+2k\pi)\) where \(k\) is any integer.

Step 2: Apply De Moivre's Theorem

De Moivre's theorem is used to find the cube roots of the complex number. According to this theorem, the cube roots of \(cis(\theta)\) are \(cis(\theta/3+2k\pi/3)\) where \(k=0, 1, 2\).

Step 3: Calculate the roots

Calculate the cube roots by substituting the appropriate value of \(k\) into the equation from step 2. The roots are \(cis(\pi/6)\), \(cis(5\pi/6)\) and \(cis(3\pi/2)\).

Step 4: Convert to Rectangular Form

The final step is to convert each root from polar form to rectangular form. This is done by using the relations \(x=cos(\theta)\) and \(y=sin(\theta)\) to give \(i/2+\sqrt{3}/2\), \(-i/2+\sqrt{3}/2\), and \(-i\).

Key Concepts

Cube Roots

Finding the cube roots of a complex number involves determining the solutions to the equation \(z^3 = i\). Cube roots of complex numbers are interesting because they exhibit rotational properties on the complex plane. Complex numbers can be represented as points on a two-dimensional plane, with one axis representing the real part and the other the imaginary part.

• To find the cube roots, we first express the complex number in polar form which gives us an easier way to compute roots.
• In the context of complex numbers, there are always 3 distinct solutions for cube roots.
• These solutions can be evenly spaced around a circle on the complex plane.

Each solution can be thought of as rotating the complex number around the origin. This geometric interpretation helps to visualize the properties of the result.

Polar Form

The polar form of a complex number is one that expresses a complex number in terms of its magnitude and angle. It's a very versatile way of representing complex numbers, especially when performing multiplication or finding roots.

• For any complex number, the polar form is expressed as \(r \text{cis}(\theta)\), where \(r\) is the magnitude \(|z|\), and \(\theta\) is the argument (angle) with respect to the real positive axis.
• This form is handy for simplifying operations like multiplication, division, and finding powers and roots.

In this exercise, the complex number \(i\) is expressed in polar form as \(\text{cis}(\pi/2)\), indicating it is placed at 90 degrees on the complex plane. The polar form is pivotal for using De Moivre's theorem effectively.

De Moivre's Theorem

De Moivre's theorem is a powerful tool in complex number theory. It simplifies the process of raising complex numbers to a power or extracting roots.

• The theorem states that for any real number \(n\), \((\text{cis}(\theta))^n = \text{cis}(n\theta)\).
• This means to find cube roots, or any roots, instead of dealing with complex arithmetic directly, you distribute the division across the angle term \(\theta\).
• In our case, for cube roots of \(i\), set \(\theta = \pi/2\), and solve for \(n\times\pi/2\).

This theorem transforms the process into a simple matter of arithmetic with angles, rather than calculus of complex numbers.

Rectangular Form

While polar form is useful for certain operations, the rectangular form shows the complex number in terms of its real and imaginary components, \(a+bi\). Converting from polar to rectangular involves using trigonometric functions.

• In this form, the complex number \(z = r(\cos\theta + i\sin\theta)\), simplifies to \(a = r\cos\theta\) and \(b = r\sin\theta\).
• For the cube roots of \(i\), turning polar results back to rectangular form gives concrete numerical values which are often easier to interpret.
• This form is typically more practical for arithmetic operations and complex number visualization in standard coordinate system terms.

Understanding both forms and how they convert is crucial in complex number calculations, providing clarity and option for interpretations.