Problem 56 - Algebra and Trigonometry

Question

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$ \left[2\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)\right]^{3} $$

Step-by-Step Solution

Answer

The rectangular form of the given complex number raised to the power of 3 is \(-4 + 4\sqrt{3}i\).

Step 1: Apply DeMoivre's Theorem

Raise the given complex number to the power of 3 using DeMoivre's theorem, which will result in \(2^{3}(\cos 3 \times 40^{\circ} + i \sin 3 \times 40^{\circ})\). Calculate the value to get \(8(\cos 120^{\circ} + i \sin 120^{\circ})\).

Step 2: Convert to Rectangular Form

Convert from trigonometric form to rectangular form. The rectangular form of a complex number is \(x + yi\) or \(r\cos\theta + r i\sin\theta\), where \(x=r\cos\theta\) and \(y=r\sin\theta\). Applying these formulas for \(r = 8\), \(\cos\theta = \cos 120^{\circ}\), and \(\sin\theta = \sin 120^{\circ}\), we get \(-4 + 4\sqrt{3}i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. You can express complex numbers in the form \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. Here, \(i\) is the imaginary unit, defined by the property that \(i^2 = -1\). This might seem daunting at first, but it's simple when you break it down.

Imagine the complex number as a point plotted on a plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This is known as the complex plane. The real component, \(a\), is how far along the horizontal (or real) axis the point is, while the imaginary component, \(b\), is how far along the vertical (or imaginary) axis the point is.

• Real part (Re): It determines the position on the horizontal axis.
• Imaginary part (Im): It decides the location on the vertical axis.

Understanding complex numbers is essential because they extend the concept of one-dimensional number lines to two dimensions, helping solve equations that do not have solutions in the realm of real numbers alone.

Rectangular Form

The rectangular form of a complex number is a representation that clearly shows the real and imaginary components as \(x + yi\). This form is perfect for adding or subtracting complex numbers because it allows you to combine like terms easily.

In the exercise solution, the final form \(-4 + 4\sqrt{3}i\) is an example of a complex number expressed in this rectangular form. Here's how the conversion from trigonometric to rectangular form works:

When converting from trigonometric to rectangular form, use the identities \(x = r\cos\theta\) and \(y = r\sin\theta\). Here, \(r\) is the modulus or absolute value of the complex number, and \(\theta\) is the argument or direction angle.

• Computation: Find \(x\) and \(y\) using the above identities. For example, from \(8(\cos 120^{\circ} + i \sin 120^{\circ})\), we calculate \(x = 8 \cos 120^{\circ} = -4\) and \(y = 8 \sin 120^{\circ} = 4\sqrt{3}\).
• Assembly: Combine \(x\) and \(y\) to express the complex number in rectangular form, demonstrating both parts visibly.

Rectangular form clearly delineates the structure of complex numbers, simplifying operations like addition and subtraction.

Trigonometric Form

Trigonometric form of a complex number is useful for multiplication, division, and finding powers of complex numbers. It expresses the number as \(r(\cos\theta + i\sin\theta)\), where \(r\) is the modulus—distance from the origin to the point in the complex plane, and \(\theta\) is the argument—angle from the positive real axis.

In trigonometric form, multiplication and raising to powers become more straightforward due to DeMoivre's theorem, which states: \((r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\).

• Modulus (\(r\)): This is calculated as the distance from the origin using \(\sqrt{a^2 + b^2}\).
• Argument (\(\theta\)): This is calculated using the inverse tangent, \(\arctan\left(\frac{b}{a}\right)\), adjusted according to the quadrant the angle lies in.

In the exercise, applying DeMoivre's theorem allowed finding the power directly in trigonometric form first, which was then converted to rectangular form. This method simplifies calculating powers of complex numbers by transforming it into an operation on the polar coordinates.