Problem 23 - Advanced Engineering Mathematics

Question

$$\left(\frac{\sqrt{2}}{2}\right)^{10}\left[\cos \frac{10 \pi}{4}+i \sin \frac{10 \pi}{4}\right]=\frac{1}{32} i$$

Step-by-Step Solution

Answer

The given expression equals \( \frac{1}{32} i \).

Step 1: Simplify the Expression Inside the Exponent

First, let's simplify the fraction \( \frac{10\pi}{4} \). Dividing the numerator and the denominator by 2, we get \( \frac{5\pi}{2} \), which can be rewritten as \( 2\pi + \frac{\pi}{2} \). Since \( \cos(\theta) \) and \( \sin(\theta) \) are periodic with period \( 2\pi \), \[ \cos \left( \frac{5\pi}{2} \right) = \cos \left( \frac{\pi}{2} \right) = 0 \]\[ \sin \left( \frac{5\pi}{2} \right) = \sin \left( \frac{\pi}{2} \right) = 1 \]

Key Concepts

Trigonometric Form of Complex Numbers

Complex numbers can be expressed in various ways, but one of the most valuable forms is the trigonometric form. This representation is perfect when you want to multiply, divide or find powers and roots of complex numbers easily. A complex number in trigonometric form is given by:\[ z = r (\cos \theta + i \sin \theta) \]- **\( r \)** represents the modulus or magnitude of the complex number.- **\( \theta \)** is the argument, which is the angle the complex number makes with the positive real axis in the complex plane.This form is particularly useful because of its connection to trigonometry and geometry. Knowing how to transition between the standard form, \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, and the trigonometric form is key to tackling complex number problems efficiently.

Polar Coordinates

The concept of polar coordinates is an extension of the traditional Cartesian coordinate system, adding a new dimension of clarity to how we view complex numbers. Instead of representing a point on the plane with \( (x, y) \), polar coordinates use \( (r, \theta) \):

• \( r \) is the distance from the origin to the point, equivalent to the modulus of a complex number.
• \( \theta \) is the angle from the positive x-axis to the line connecting the origin to the point, akin to the argument of a complex number.

This approach does not only help in graphing complex numbers but also simplifies calculations involving powers and roots.
Using polar coordinates, complex numbers become easier to manipulate geometrically and computationally. Therefore, understanding polar coordinates offers a solid foundation for moving forward with the deeper properties of complex numbers.

Euler's Formula

Euler's formula is a beautiful and elegant bridge between the exponential function and trigonometric functions. It states:\[ e^{i\theta} = \cos \theta + i \sin \theta \]This relation allows us to represent complex numbers in the exponential form \( z = re^{i\theta} \), which is incredibly powerful in calculations.

• It highlights the cyclical nature of the exponential function when dealing with complex exponents.
• It shows why complex numbers on a unit circle can be so easily expressed in this tidy form.

Using Euler's formula simplifies operations on complex numbers, especially when raising them to powers, as the operation simplifies to multiplying exponents. This exponential form is rooted deeply in both mathematics and engineering, making many processes and calculations much more intuitive.

De Moivre's Theorem

De Moivre's Theorem is an excellent tool for finding powers and extracting roots of complex numbers. It can be stated as:\[ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \]For a complex number in trigonometric form, this theorem allows easy calculation of powers by just multiplying the argument, \( \theta \), by the power, \( n \).
This greatly simplifies many problems, including those like the original exercise where a complex number is raised to the power of 10. Simply applying De Moivre can lead directly to a solution without lengthy calculations.

• It's efficient for high powers and can be further extended for fractional powers in roots of complex numbers.
• It simplifies finding new complex numbers derived from a given one.

Understanding how to apply De Moivre's Theorem offers a straightforward way to navigate through complex number manipulations with precision and ease.