Problem 25 - Advanced Engineering Mathematics

Question

In Problems 21-26, use (8) to compute the indicated power. $$ \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)^{12} $$

Step-by-Step Solution

Answer

The result is \(-i\).

Step 1: Identify the formula

The given expression is in the form of the complex number in polar form, \( \cos \theta + i \sin \theta \). These types of problems are often solved using De Moivre's Theorem, which in formula form is \( \left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta) \).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, we can rewrite the given expression \( \left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right)^{12} \) as \( \cos \left( 12 \times \frac{\pi}{8} \right) + i \sin \left( 12 \times \frac{\pi}{8} \right) \).

Step 3: Simplify the multiplication inside the trigonometric functions

Calculate \( 12 \times \frac{\pi}{8} \):\[12 \times \frac{\pi}{8} = \frac{12\pi}{8} = \frac{3\pi}{2}.\]

Step 4: Find the trigonometric values

Now substitute \( \frac{3\pi}{2} \) into the trigonometric functions:\[ \cos \left( \frac{3\pi}{2} \right) = 0 \] and \[ \sin \left( \frac{3\pi}{2} \right) = -1. \]

Step 5: Write the final result

Using the values obtained, the expression \( \cos \left( \frac{3\pi}{2} \right) + i\sin \left( \frac{3\pi}{2} \right) \) becomes:\[ 0 + i(-1) = -i. \]

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for working with complex numbers, especially when they are in polar form. The theorem is expressed as:

• \( \left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta) \).

In simple terms, it allows us to raise a complex number in polar form to any integer power easily. The main idea is to multiply the angle \(\theta\) by the power \(n\), while the modulus (or the magnitude) of the complex number is raised to the same power if it is given explicitly.

This theorem simplifies complex multiplication tasks and can also be used for finding roots of complex numbers. For example, in the given problem, the expression \( \left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right)^{12} \) can be evaluated quickly by calculating \(12 \times \frac{\pi}{8} = \frac{3\pi}{2} \), then finding the cosine and sine of the resulting angle.

Polar Form

The polar form of a complex number is an alternative way of expressing complex numbers using trigonometry. Instead of using the standard Cartesian form \(a + bi\), a complex number can be expressed as:

• \(r(\cos \theta + i \sin \theta)\),

where \(r\) is the modulus of the complex number, equal to the distance from the origin to the point \((a, b)\) in the complex plane. \(\theta\) is the argument, which is the angle made with the positive real axis.

This form is particularly useful when multiplying or dividing complex numbers, raising them to powers, or finding roots, as it simplifies trigonometric calculations. For instance, raising a complex number to the 12th power in polar form, as seen in the exercise, involves just multiplying the angle \(\theta\) by 12. This reduces the computation significantly compared to the Cartesian form.
Understanding how to convert between Cartesian and polar forms is crucial for fully mastering these operations with complex numbers.

Trigonometric Functions

Trigonometric functions, namely sine and cosine, play an essential role in expressing complex numbers in polar form. They represent the coordinates of points on the unit circle.

• \(\cos \theta\) denotes the x-coordinate,
• \(i \sin \theta\) denotes the y-coordinate where \(i\) is the imaginary unit.

These functions help define the position of complex numbers in the Argand plane.

In the context of the exercise, trigonometric functions help in breaking down the original expression into a form that can be easily manipulated using De Moivre's theorem. When \( \frac{3\pi}{2} \) is calculated, it determines the x and y coordinates on the unit circle, where \( \cos \left( \frac{3\pi}{2} \right) = 0 \) and \( \sin \left( \frac{3\pi}{2} \right) = -1 \).
Through these, the trigonometric identities and relationships simplify the calculation process and transform complex operations into manageable tasks. Mastery of trigonometric functions provides deeper understanding and efficiency in solving complex number problems.