Problem 82 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left[2\left(\cos\ \dfrac{\pi}{8}\ +\ i\ \sin\ \dfrac{\pi}{8}\right)\right]^6\)

Step-by-Step Solution

Answer

The sixth power of the complex number \(2(\cos \dfrac{\pi}{8} + i \sin \dfrac{\pi}{8})\) is \(-32\sqrt{2} + i 32\sqrt{2}\).

Step 1: Applying DeMoivre's Theorem

To find the sixth power of the complex number \[2(\cos \dfrac{\pi}{8} + i \sin \dfrac{\pi}{8})\], apply DeMoivre's theorem directly as follows: \[ \bigg[2\left(\cos\ \dfrac{\pi}{8}\ +\ i\ \sin\ \dfrac{\pi}{8}\right)\bigg]^6 = 2^6\left(\cos\ \dfrac{6\pi}{8}\ +\ i\ \sin\ \dfrac{6\pi}{8}\right)\]. This simplifies to \[64\left(\cos\ \dfrac{3\pi}{4}\ +\ i\ \sin\ \dfrac{3\pi}{4}\right)\]

Step 2: Computing cos and sin values

Next we can substitute the familiar trigonometric values \(\cos \left(\frac{3\pi}{4}\right) = - \frac{\sqrt{2}}{2}\) and \(\sin \left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\) to get: \[64\left(-\frac{\sqrt{2}}{2} +\ i \frac{\sqrt{2}}{2}\right)\]

Step 3: Transforming to Standard Form

Transforming the complex number to standard form (a + bi), we can simplify the last step as: \[ -32\sqrt{2} + i 32\sqrt{2}\]

Key Concepts

Complex Numbers

Complex numbers extend our familiar number system. They consist of two parts: a real component and an imaginary component. The real part is similar to our usual numbers. The imaginary part is a multiple of the imaginary unit, denoted as \(i\), where \(i\) is defined such that \(i^2 = -1\).

• A complex number is often written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
• The beauty of complex numbers is in how they allow us to solve equations that have no solutions in the real number system.

In our exercise, we started with the complex number in trigonometric form \(2(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8})\). This way of expressing complex numbers is very useful for calculations involving powers and roots due to DeMoivre's Theorem.

Trigonometry

Trigonometry helps us understand and calculate relationships between the angles and sides of triangles. It becomes fundamental in working with complex numbers in polar form.

• In trigonometry, \(\cos\theta\) represents the horizontal component of a vector at angle \(\theta\) to a reference line.
• Similarly, \(\sin\theta\) represents the vertical component.

For complex numbers expressed as \(r(\cos\theta + i\sin\theta)\), DeMoivre's Theorem utilizes these trigonometric functions to simplify calculations of powers. In our step-by-step solution, trigonometric values like \(\cos \frac{3\pi}{4} = - \frac{\sqrt{2}}{2}\) and \(\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}\) were essential in transforming our complex number into a manageable form.

Standard Form

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. This format is straightforward and shows both the real and imaginary parts clearly.

• When converting from trigonometric form to standard form, it often requires calculating specific trigonometric values.
• The result is simplified in the form \(a + bi\), clarifying the magnitude and direction of the number in a two-dimensional plane.

In our example, through applying necessary trigonometric identities and simplifying, the final step was to express the complex number as \(-32\sqrt{2} + i 32\sqrt{2}\). This form makes it more accessible for further computation or interpretation.