Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{5}$$

Step-by-Step Solution

Answer

The power is \(-1\).

Step 1: Identify the Form

The expression \((\cos \frac{\pi}{5} + i \sin \frac{\pi}{5})\) is written in trigonometric form for a complex number. This is equivalent to \(\text{cis}\left(\frac{\pi}{5}\right)\).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that \((\text{cis}\, \theta)^n = \text{cis}\, (n\theta)\). For this problem, set \(\theta = \frac{\pi}{5}\) and \(n = 5\). Thus, \((\cos \frac{\pi}{5} + i \sin \frac{\pi}{5})^5 = \text{cis}\left(5 \times \frac{\pi}{5}\right) = \text{cis}(\pi)\).

Step 3: Calculate the Rectangular Form of \(\text{cis}(\pi)\)

The expression \(\text{cis}(\pi)\) means \(\cos \pi + i \sin \pi\). We know that \(\cos \pi = -1\) and \(\sin \pi = 0\). Therefore, \(\text{cis}(\pi)\) in rectangular form is \(-1 + 0i\), which simplifies to \(-1\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a fundamental tool in complex number arithmetic, particularly useful for raising complex numbers to powers. Its formula is expressed as \((\text{cis}\, \theta)^n = \text{cis}\, (n\theta)\). Here, \(\theta\) is the angle, and \(n\) is the power to which the complex number is raised. This theorem simplifies the computation of powers of complex numbers when they are in trigonometric form.

• To apply the theorem, identify the angle \(\theta\) and the power \(n\).
• Compute the resulting angle by multiplying \(\theta\) by \(n\).
• Express the result in trigonometric form: \(\text{cis}(n\theta)\).

In the given problem, \(\theta = \frac{\pi}{5}\) and \(n = 5\). Using De Moivre's Theorem, the expression becomes \(\text{cis}(\pi)\). This approach makes it easy to handle powers of complex numbers without extensive calculation.

Trigonometric Form

The trigonometric form of a complex number, also known as polar form, expresses a complex number in terms of its magnitude and angle. It is typically written as \(r(\cos\theta + i\sin\theta)\) or \(r\text{cis}\theta\), where \(r\) is the magnitude and \(\theta\) is the angle.

This form is advantageous when performing operations like multiplication, division, and exponentiation of complex numbers. It transforms complicated algebraic manipulations into simple operations with angles and magnitudes.

Key points include:

• The magnitude \(r\) is the distance from the origin to the point in the complex plane.
• The angle \(\theta\), measured from the positive x-axis, defines the direction of the number.
• The trigonometric form leverages Euler's formula to relate trigonometric functions to exponential form, \(e^{i\theta} = \cos\theta + i\sin\theta\).

In the exercise, the complex number \((\cos \frac{\pi}{5} + i \sin \frac{\pi}{5})\) is given in trigonometric form, making it ideal for applying De Moivre's Theorem.

Rectangular Form

Rectangular, or Cartesian, form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form is intuitive as it directly corresponds to coordinates on a two-dimensional plane.

The conversion between trigonometric and rectangular forms involves calculating the cosine and sine of the angle \(\theta\), and multiplying by the magnitude \(r\).

Here's how to convert a trigonometric form to rectangular form:

• Use \(a = r \cos\theta\) to find the real part.
• Use \(b = r \sin\theta\) to find the imaginary part.
• Combine them to get \(a + bi\).

In the solution at hand, the expression \(\text{cis}(\pi)\) converts to \(\cos(\pi) + i\sin(\pi)\), which becomes \(-1 + 0i\). Therefore, the rectangular form is simply \(-1\). Understanding both forms is essential for solving complex number problems effectively.