Problem 79 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4}$$

Step-by-Step Solution

Answer

The result in standard form is \( 40.5 + 70.3i \)

Step 1: Apply DeMoivre's Theorem

Use DeMoivre's theorem \( (r(\cos \theta + i \sin \theta))^n = r^n(\cos n\theta + i \sin n\theta) \) where \( r=3 \), \( \theta = 15° \), and \( n=4 \).

Step 2: Compute the values

Calculate \( r^n \), \( \cos n\theta \) and \( \sin n\theta \). They are \( 3^4 = 81 \), \( \cos 60° \), and \( \sin 60° \) respectively.

Step 3: Express in cartesian form

Rewrite the obtained complex number in standard form, i.e., \( a + bi \). The standard values of \( \cos 60° \) and \( \sin 60° \) are \( \frac{1}{2} \) and \( \frac{\sqrt{3}}{2} \) respectively. Thus, the number in standard form is \( 81 \times \frac{1}{2} + 81 \times \frac{\sqrt{3}}{2}i = 40.5 + 70.3i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the standard form as

• \( a + bi \)

where \( a \) is the real component, and \( b \) is the imaginary component.

• The imaginary unit \( i \) follows the rule \( i^2 = -1 \).

This property allows complex numbers to extend the scope of real numbers, enabling the solution of equations that do not have real solutions.
To handle such numbers easily, especially when dealing with powers and roots, it's often beneficial to convert them to a different form. One popular form is the trigonometric or polar form of a complex number.

Trigonometric Form

The trigonometric or polar form of a complex number rewires the number in terms of its magnitude and angle, making some operations more straightforward. Any complex number \( a + bi \) can be rewritten as:

• \( r(\cos \theta + i \sin \theta) \)
• where \( r = \sqrt{a^2 + b^2} \) is the magnitude (or modulus) of the complex number.
• \( \theta = \tan^{-1}(\frac{b}{a}) \) is the argument or angle of the complex number.

Using this form simplifies multiplication and division of complex numbers, and it's particularly useful for raising complex numbers to a power. By leveraging DeMoivre's Theorem, computations that would otherwise be tedious become manageable.

Powers of Complex Numbers

DeMoivre’s Theorem is a powerful tool for finding the powers of complex numbers when they are expressed in trigonometric form. If a complex number is in the form \( r(\cos \theta + i \sin \theta) \), then raising it to the power of \( n \) is achieved by:

• Taking the radius \( r \) to the power \( n \), resulting in \( r^n \).
• Multiplying the angle \( \theta \) by \( n \) to get \( n\theta \).

Thus, \[(r(\cos \theta + i \sin \theta))^n = r^n (\cos (n\theta) + i \sin (n\theta))\]After calculating the power, the result can be converted back to the standard form \( a + bi \), which often provides more intuitive understanding and practicality in real-world applications. This conversion involves using the trigonometric values calculated to express the complex number in terms of its real and imaginary components.