Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{8}$$

Step-by-Step Solution

Answer

The rectangular form of the expression is 1.

Step 1: Identify the complex number and angle

The expression given is \( \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)^8 \). Here, the complex number is in polar form, corresponding to an angle \( \frac{\pi}{4} \). In polar form, this represents a complex number with magnitude 1 and angle \( \frac{\pi}{4} \).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that \( \left( \cos \theta + i \sin \theta \right)^n = \cos (n\theta) + i \sin (n\theta) \). For this problem, \( n = 8 \) and \( \theta = \frac{\pi}{4} \). So, we calculate \( \cos(8 \times \frac{\pi}{4}) + i \sin (8 \times \frac{\pi}{4}) \).

Step 3: Simplify the angle

Compute the angle \( 8 \times \frac{\pi}{4} = 2\pi \). The cosine and sine functions have periods of \( 2\pi \), meaning after a rotation of \( 2\pi \), we are back at the starting point. Therefore, \( \cos(2 \pi) = 1 \) and \( \sin(2 \pi) = 0 \).

Step 4: Write the rectangular form of the result

Since \( \cos(2\pi) = 1 \) and \( \sin(2\pi) = 0 \), the rectangular form of the result is simply \( 1 + 0i = 1 \).

Key Concepts

Rectangular Form

Complex numbers can be expressed in different ways. One of the most common expressions is in rectangular form. This form combines the real and imaginary parts of a complex number, allowing it to be easily plotted on a standard coordinate plane.

The general representation for a complex number in rectangular form is:

• Real part: usually denoted as \( a \)
• Imaginary part: usually denoted as \( b \)

Together, a complex number can be written as \( a + bi \), where \( i \) is the imaginary unit \((i^2 = -1)\). Rectangular form is particularly useful for adding or subtracting complex numbers, as it allows you to handle the real and imaginary components directly.

Complex Numbers

Complex numbers are an essential concept in mathematics, extending the idea of one-dimensional numbers to a two-dimensional plane. They are formed by a real part and an imaginary part.

A complex number has the form \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the imaginary part.

This allows for operations beyond what is possible with standard real numbers. Complex numbers are particularly valuable in fields requiring calculations of electrical currents, signal processing, and other applications that involve oscillation and waveforms.

Polar Form

The polar form of a complex number provides another useful way to express it, especially valuable when multiplying or dividing complex numbers. In polar form, a complex number is described by its magnitude and angle relative to the positive x-axis.

The conversion from rectangular form to polar form involves:

• Magnitude, \( r \): calculated using \( r = \sqrt{a^2 + b^2} \)
• Angle (also called argument), \( \theta \): found using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

A complex number in polar form can then be rewritten as \( r(\cos \theta + i \sin \theta) \). This approach simplifies many operations, such as exponentiation and finding roots, through the use of De Moivre's Theorem.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental to connecting geometric angles to complex numbers. These functions, defined based on angles, allow for the transformation of complex numbers from polar to rectangular form.

Key trigonometric functions involved with complex numbers include:

• Sine, \( \sin \theta \): This function gives the opposite side over the hypotenuse in a right triangle.
• Cosine, \( \cos \theta \): This function gives the adjacent side over the hypotenuse.

Utilizing these functions, any complex number in polar form \( (r\cos \theta + ir\sin \theta) \) can be shifted to rectangular form, making these equations versatile tools in various mathematical disciplines.