Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Find all indicated roots and express them in rectangular form. Check your results with a calculator. The cube roots of \(27\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)\).

Step-by-Step Solution

Answer

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

Step 1: Convert to Polar Form

The expression given is already in polar form as:\[27(\cos 180^{\circ} + i\sin 180^{\circ})\]Here, the modulus is 27, and the argument is \(180^{\circ}\).

Step 2: Find the Cube Root

To find the cube roots, we use the formula for finding the roots of a complex number:\[z^{\frac{1}{3}} = r^{\frac{1}{3}} \left( \cos \frac{\theta + 360^{\circ}k}{n} + i \sin \frac{\theta + 360^{\circ}k}{n} \right)\]where \(r=27\), \(\theta=180^{\circ}\), \(n=3\) (since we are taking the cube root), and \(k=0,1,2\).

Step 3: Calculate Each Root

Calculate each of the cube roots:For \(k=0\):- Modulus is \(r^{\frac{1}{3}} = 27^{\frac{1}{3}} = 3\).- Argument is \(\frac{180^{\circ} + 360^{\circ} \times 0}{3} = 60^{\circ}\).- So the first root is \(3 \left( \cos 60^{\circ} + i \sin 60^{\circ} \right) = 3\left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = \frac{3}{2} + i\frac{3\sqrt{3}}{2}\).For \(k=1\):- Argument is \(\frac{180^{\circ} + 360^{\circ} \times 1}{3} = 180^{\circ}\).- So the second root is \(3 \left( \cos 180^{\circ} + i \sin 180^{\circ} \right) = 3(-1) = -3\).For \(k=2\):- Argument is \(\frac{180^{\circ} + 360^{\circ} \times 2}{3} = 300^{\circ}\).- So the third root is \(3 \left( \cos 300^{\circ} + i \sin 300^{\circ} \right) = 3\left(\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = \frac{3}{2} - i\frac{3\sqrt{3}}{2}\).

Step 4: Express in Rectangular Form

The cube roots in rectangular form are:- \(\frac{3}{2} + i\frac{3\sqrt{3}}{2}\)- \(-3\)- \(\frac{3}{2} - i\frac{3\sqrt{3}}{2}\)

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This field of mathematics allows us to deal with a whole new set of numbers that can solve equations involving square roots of negative numbers.
In the context of cube roots, we often deal with complex numbers as they express rotations and scaling in a more comprehensive way. Complex numbers that are not solved fully in the rectangular form can lead to a rich understanding when represented in the polar or exponential forms.

Polar Form

The polar form of a complex number expresses it in terms of a magnitude (or modulus) and an angle (or argument). For example, the complex number \(z = 27(\cos 180^{\circ} + i\sin 180^{\circ})\) is represented where 27 is the modulus and \(180^{\circ}\) is the argument.

• Modulus: The modulus \(r\) is the distance from the origin to the point in the complex plane, which can be calculated using the formula \(r = \sqrt{a^2 + b^2}\).
• Argument: The argument \(\theta\) is the angle formed with the positive real axis, found using \(\theta = \arctan(\frac{b}{a})\).

This form is particularly useful when performing operations like multiplication, division, or finding powers and roots of complex numbers. You'll find it much easier to visualize rotations and transformations using polar coordinates.

Rectangular Form

The rectangular form of a complex number is its expression like \(a + bi\), where \(a\) is the real part and \(b\) the imaginary part. For our cube roots of the original problem, this form is essential for simplifying and checking the results.
After computing the roots in polar form, converting them to rectangular form involves evaluating the cosine and sine components. This transformation enables clear comparison and understanding in practical applications, like plotting or simplifying further calculations.

• Example: After computing one of the cube roots, \(3 (\cos 60^{\circ} + i \sin 60^{\circ})\), you transform this to \(\frac{3}{2} + i\frac{3\sqrt{3}}{2}\) by calculating the cosine and sine values.

Switching between forms can help simplify complex arithmetic operations, especially when combined with graphic tools like a calculator.

Roots of Complex Numbers

Finding roots, especially cube roots, of complex numbers requires understanding of various forms of these numbers and a clear formula:\[z^{\frac{1}{n}} = r^{\frac{1}{n}} \left( \cos \frac{\theta + 360^{\circ}k}{n} + i \sin \frac{\theta + 360^{\circ}k}{n} \right)\] where \(r\) is the modulus, \(\theta\) is the argument, \(n\) is the degree of the root, and \(k\) is an integer from 0 to \(n-1\).

• For each integer \(k\), you rotate \(360^{\circ}k/n\), giving different orientations of the root in the complex plane.
• This gives distinct roots that offer a symmetric pattern.

After determining these roots in polar form, converting them to rectangular form allows for practical interpretation and validation. In exercises, always compare your analytical results with graphical or calculator checks to build a solid understanding of these concepts.