Problem 58 - 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. $$ \left[\frac{1}{2}\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)\right]^{5} $$

Step-by-Step Solution

Answer

The 5th power of the given complex number in rectangular form is \( \frac{i}{32} \)

Step 1: Identify the terms

Identify the modulus \( r \), the argument \( \phi \) and the power \( n \). In the given problem, \( r \) is \( \frac{1}{2} \), \( \phi \) is \( \frac{\pi}{10} \), and \( n \) is 5.

Step 2: Apply DeMoivre's Theorem

Apply DeMoivre's theorem \[ (r(\cos \phi + i\sin \phi))^n = r^n(\cos n\phi + i\sin n\phi) \] to find the 5th power of the given complex number. Substituting the respective values we get: \[ \left( \frac{1}{2} \right)^5 (\cos 5 \cdot \frac{\pi}{10} + i \sin 5 \cdot \frac{\pi}{10}) \]

Step 3: Simplify the expression

Now simplify the expression. It gives \[ \frac{1}{32} (\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) \]

Step 4: Convert to Rectangular Form

The last step is to convert the expression back into rectangular form. Since \( \cos \frac{\pi}{2} \) is 0 and \( \sin \frac{\pi}{2} \) is 1, upon substituting we get \[ 0 + \frac{i}{32} \] Therefore, the answer in rectangular form is \( \frac{i}{32} \).

Key Concepts

Complex Numbers

Complex numbers are a fundamental aspect of mathematics, encompassing a much broader range of values than the familiar real numbers. The square roots of negative numbers are not defined within the realm of real numbers, which is precisely the gap that complex numbers fill. A complex number is expressed in the form of
\[\begin{equation}a + bi\br\br\br\br\br\br\br\end{equation}\]where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit satisfying the condition that \br\br\br\br\[\begin{equation}i^2 = -1\br\br\br\br\br\br\br\end{equation}\]This rich number system is essential for various fields such as engineering, physics, and many other scientific endeavors where computations extend beyond the real number boundary.

Rectangular Form

The rectangular form of a complex number is essentially its algebraic representation on the complex plane. It correlates with the coordinates of a point on a two-dimensional plane. In rectangular form, a complex number is represented as a combination of its real and imaginary components—as \br\br\br\br\[\begin{equation}a + bi\br\br\br\br\br\br\br\end{equation}\]Where 'a' represents the x-coordinate (real axis) and 'b' represents the y-coordinate (imaginary axis). This form is highly practical when adding or subtracting complex numbers because it involves a simple combination of these two components. Rectangular form is especially useful when dealing with problems that require the visual interpretation of a number's position or when analyzing circuits in electrical engineering.

Powers of Complex Numbers

Raising complex numbers to powers is an operation that can be greatly simplified by using DeMoivre's Theorem. The theorem is a powerful tool for computing powers of complex numbers where you have a complex number in polar form, \br\br\br\br\[\begin{equation}r (\text{{cos}} \theta + i\text{{sin}} \theta)\br\br\br\br\br\br\br\end{equation}\]and you want to raise it to an integer power 'n'. DeMoivre's Theorem states that: \br\br\br\br\[\begin{equation}(r (\text{{cos}} \theta + i\text{{sin}} \theta))^n = r^n (\text{{cos}} (n\theta) + i\text{{sin}} (n\theta))\br\br\br\br\br\br\br\end{equation}\]This simplifies the process by allowing the exponent to be applied directly to the modulus and to the angle (argument), conveniently calculated on the unit circle. After raising to a power, it's common to convert the result back to rectangular form for the final answer, which is straightforward if the argument is a multiple of \br\br\br\br\[\begin{equation}\frac{\theta}{4}\br\br\br\br\br\br\br\end{equation}\]since the trigonometric functions of common angles are widely known. The ability to convert between forms and use DeMoivre's theorem effectively is an indispensable skill in complex analysis and related disciplines.