Problem 121 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)\right]^{2}$$

Step-by-Step Solution

Answer

The power of the complex number in standard form is \( \frac{9 \sqrt{2}}{2} + i \frac{9 \sqrt{2}}{2}\).

Step 1: Apply DeMoivre's Theorem

The task asks to calculate the square of the given complex number. DeMoivre's Theorem states that \((r(\cos θ+i\sin θ))^n = r^n (\cos nθ + i \sin nθ)\), where \(r\) is the modulus of the complex number, \(θ\) is the argument of the complex number, and \(n\) is the power to which the complex number is raised. Apply this theorem to find \( [3(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8})]^2 = 3^2 (\cos 2\frac{\pi}{8} + i \sin 2\frac{\pi}{8}) \).

Step 2: Simplify the expression

Simplify the right-hand side to get \( 9(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) \). The cosine of \(\frac{\pi}{4}\) and the sine of \(\frac{\pi}{4}\) are both \(\frac{1}{\sqrt{2}}\). Substituting these values into the expression yields \(9(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}})\).

Step 3: Convert to standard form

Multiply through by the real number 9 to get \(9*\frac{1}{\sqrt{2}} + 9 *i \frac{1}{\sqrt{2}}\). Simplifying this gives \( \frac{9}{\sqrt{2}} + i \frac{9}{\sqrt{2}}\), or converting radical in the denominator, the result is \( \frac{9 \sqrt{2}}{2} + i \frac{9 \sqrt{2}}{2}\). This is the standard form of the complex number.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are generally written in the form \( a + bi \). Here, \( a \) is the real part, and \( bi \) is the imaginary part, with \( i \) being the imaginary unit, equal to the square root of \(-1\).
These numbers allow mathematicians and engineers to solve equations that have no real solutions. For example, the equation \( x^2 + 1 = 0 \) has no real solution, but has complex solutions \( x = i \) and \( x = -i \).

• They are used in various fields, such as engineering, physics, and applied mathematics.
• Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, though special rules apply when dealing with the imaginary unit \(i\).

Understanding complex numbers is crucial when working with DeMoivre's Theorem, which helps calculate powers and roots of complex numbers.

Modulus and Argument

To work with complex numbers in polar form, we need to know two things: the modulus and the argument.
The modulus of a complex number is its 'size' or 'magnitude'. It is the distance from the origin to the point \( (a, b) \) on the complex plane, and is calculated using the Pythagorean theorem: \( r = \sqrt{a^2 + b^2} \). This is the radius \( r \) in polar form.

• The argument of a complex number is the angle \( \theta \), measured in radians, counterclockwise from the positive x-axis to the line connecting the origin with \( (a, b) \).
• It is found using the formula \( \theta = \tan^{-1}(\frac{b}{a}) \).

Understanding these two components allows us to use DeMoivre's Theorem and easily convert complex numbers between rectangular and polar forms. This simplifies operations like exponentiation significantly.

Standard Form of Complex Numbers

The standard form of complex numbers is often referred to as their rectangular form, expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
Writing complex numbers in standard form is essential when performing arithmetic operations and when interpreting the results of calculations, such as those involving DeMoivre's Theorem.

• When converting from polar to rectangular form, we use the expressions \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \).
• This conversion makes it easier to visualize complex numbers and understand the combination of their real and imaginary components.

In practical problems, once the operations are completed in polar form, the results are often converted back to the standard form to provide a clear, comprehensible final answer.