Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)^{15} $$

Step-by-Step Solution

Answer

The complex number in the form \( a + bi \) is \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \).

Step 1: Convert to Polar Form

The given complex number is \(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i\). First, identify the modulus \(r\) using \( r = \sqrt{x^2 + y^2} \), where \( x = -\frac{\sqrt{2}}{2} \) and \( y = \frac{\sqrt{2}}{2} \). Calculating this gives \( r = 1 \). Next, find the angle \( \theta \) using \( \tan(\theta) = \frac{y}{x} = -1\), which implies \( \theta = \frac{3\pi}{4} \). Thus, the polar form is \(1\left( \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right)\).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states \((r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))\). Here, \( n = 15 \). So, we have \(1^{15}(\cos(15\times\frac{3\pi}{4}) + i\sin(15\times\frac{3\pi}{4})) = \cos\left(\frac{45\pi}{4}\right) + i\sin\left(\frac{45\pi}{4}\right)\).

Step 3: Simplify the Angles

Compute \( \frac{45\pi}{4} \) to find its equivalent angle within a single revolution \([0, 2\pi)\). \( \frac{45\pi}{4} \) is equivalent to \(\frac{45\pi}{4} - 11\pi = \frac{\pi}{4}\) since two full revolutions is \(8\pi\). Thus, the angle is \(\frac{\pi}{4}\).

Step 4: Determine the Trigonometric Values

From \( \cos\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{4}\right)\), we know \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) and \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Thus, the expression returns to rectangular form as \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \).

Key Concepts

Complex Numbers

Complex numbers are a crucial concept in mathematics and engineering due to their comprehensive way of representing various phenomena. A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The unit \(i\) is the imaginary unit defined by the property \(i^2 = -1\). Complex numbers extend the idea of one-dimensional number systems, like real numbers, to a two-dimensional one, which notably includes solving equations that don't have real solutions, such as \(x^2 + 1 = 0\). In the complex plane, you can visualize complex numbers as points or vectors with the real part on the x-axis and the imaginary part on the y-axis. Understanding them is foundational for more advanced areas of study like signal processing and quantum mechanics.

Polar Form

The polar form of a complex number provides an alternative representation that is particularly useful for multiplication and powers. A complex number can be transformed into polar form \(r(\cos\theta + i\sin\theta)\), where \(r\) is the modulus and \(\theta\) is the argument or angle.

• The modulus, \(r\), represents the distance of the complex number from the origin of the complex plane and can be found using \(r = \sqrt{a^2 + b^2}\).
• The argument, \(\theta\), is the angle made with the positive real axis and is calculated using trigonometric functions.

This form is instrumental when using De Moivre's Theorem, which simplifies the process of raising complex numbers to powers. Polar coordinates make it straightforward to perform operations that would be cumbersome in standard rectangular form.

Trigonometric Functions

Trigonometric functions like cosine and sine are extensively used to describe the components of a complex number in polar form. These functions map an angle \(\theta\) to the unit circle, helping transition between polar and rectangular forms seamlessly.cosine, \(\cos(\theta)\), gives the horizontal component, while sine, \(\sin(\theta)\), provides the vertical component of a complex number. When a complex number is expressed as \(r(\cos\theta + i\sin\theta)\), these functions determine its exact position or vector representation in the complex plane.Understanding these trigonometric functions and their properties is essential for accurately solving problems involving rotations, oscillations, and periodic phenomena, making the mastery of this concept a significant advantage in various scientific and engineering fields.

Rectangular Form

The rectangular form of a complex number, often considered the standard form, represents a complex number as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form directly relates to our everyday view of numbers and extends it by including the imaginary number \(i\).Rectangular coordinates are straightforward for addition and subtraction of complex numbers. However, operations like multiplication, division, and exponentiation often become more manageable when the number is converted to polar form. Once the operation is complete, the result can be returned to rectangular form by finding the trigonometric values of \(\cos\theta\) and \(\sin\theta\) and applying them back to \(a\) and \(b\).Having a good grasp of switching between these two forms allows for flexibility and a deeper understanding of complex number manipulations, particularly for engineering and physics calculations where both forms are frequently utilized.