Problem 56 - Algebra and Trigonometry

Question

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ (1-\sqrt{5} i)^{8} $$

Step-by-Step Solution

Answer

(1-\sqrt{5}i)^8 = 1296 e^{-8i\tan^{-1}(\sqrt{5})}

Step 1: Identify Polar Coordinates

First, convert the complex number to polar form. Given the complex number is \(1-\sqrt{5}i\), identify \(r\) and \(\theta\).

Step 2: Calculate Magnitude

Calculate the magnitude \(r\) using the formula \(r = \sqrt{x^2 + y^2}\). For \(1-\sqrt{5}i\): \[r = \sqrt{1^2 + (-\sqrt{5})^2} = \sqrt{1 + 5} = \sqrt{6}.\]

Step 3: Calculate Argument

Calculate the argument \(\theta\) using \(\tan(\theta) = \frac{y}{x}\). Here, \(x=1\) and \(y=-\sqrt{5}\), giving: \[\tan(\theta) = \frac{-\sqrt{5}}{1} = -\sqrt{5}. \]

Step 4: Find Angle \(\theta\)

Determine the correct quadrant and find \(\theta\). Since the complex number is in the fourth quadrant (positive real part, negative imaginary part), calculate: \[\theta = -\tan^{-1}(\sqrt{5}). \]

Step 5: Convert to Exponential Form

Using the values of \(r\) and \(\theta\), the exponential form is: \[(1 - \sqrt{5}i) = \sqrt{6} e^{-i\tan^{-1}(\sqrt{5})}.\]

Step 6: Apply De Moivre's Theorem

Apply De Moivre's theorem to find \((\sqrt{6} e^{-i\tan^{-1}(\sqrt{5})})^8\): \[r^n e^{in\theta} = (\sqrt{6})^8 e^{-8i\tan^{-1}(\sqrt{5})}.\]

Step 7: Simplify Magnitude and Argument

Simplify the magnitude and argument: \[(\sqrt{6})^8 = 6^4 = 1296\] and \[-8\tan^{-1}(\sqrt{5}).\]

Step 8: Convert Back to Rectangular Form

To convert back, use \[e^{i\theta} = \cos(\theta) + i \sin(\theta)\]. Computational methods yield numeric values for \(\cos\) and \(\sin\). Hence, the rectangular form is not easily placed visually, but follows the values derived.

Key Concepts

polar coordinates

Polar coordinates provide an alternative way to represent complex numbers. Instead of using the standard rectangular form (real and imaginary parts), they use a magnitude (r) and an argument (θ). The magnitude measures the distance from the origin to the point in the complex plane. The argument measures the angle formed with the positive real axis.
To convert a complex number in polar coordinates to rectangular form, you can use the connections:

• Real part (x) = r * cos(θ)
• Imaginary part (y) = r * sin(θ)

By mastering polar coordinates, you will gain a deeper insight into the geometry of complex numbers.

rectangular form

The rectangular form represents complex numbers as an addition of real and imaginary parts. It is especially useful for arithmetic operations and visualization in the complex plane.
A complex number in rectangular form is written as:
\(x + yi\), where:

• \(x\): the real part
• \(y\): the imaginary part

For any complex number in polar form, we can convert it to rectangular form using:

• \(x = r * cos(\theta)\)
• \(y = r * sin(\theta)\)
  

Rectangular forms provide an easy way to perform addition and subtraction of complex numbers.

exponential form

The exponential form leverages Euler's formula:
\(e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta)\)
A complex number is represented as:

• r * \(e^{i\theta}\)

where

• \(r\) is the magnitude
• \(θ\) is the argument

Exponential form is compact and convenient for multiplication and division of complex numbers. To convert back to rectangular form:

• Real part = \(r * \text{cos}(\theta)\)
• Imaginary part = \(r * \text{sin}(\theta)\)
/ul>Using exponential form simplifies the use of powerful theorems like De Moivre's theorem.

De Moivre's theorem

De Moivre's theorem is crucial for working with powers and roots of complex numbers. It states:
If z = \(r e^{i\theta}\), then \(z^n = r^n e^{i n \theta}\).
This powerful theorem allows us to raise a complex number to any integer power easily.
For instance, given:\(z = 1 - \text{sqrt}{5}i\), in polar form it converts to:
\(z = \text{sqrt}{6} e^{-i \text{tan}^{-1}(\text{sqrt}{5})}\)Using De Moivre's theorem, raising \(z\) to the power of 8 becomes:
\((\text{sqrt}{6})^8 e^{-8i \text{tan}^{-1}(\text{sqrt}{5})}\) = \(1296 e^{-8i \text{tan}^{-1} \text{sqrt}{5}}\)

magnitude and argument

The magnitude and argument are foundational to understanding polar and exponential forms.

• Magnitude (\(r\)): the distance from the origin to the point, calculated as \(r = \text{sqrt}(x^2 + y^2)\).
• Argument (\(θ\)): the angle formed with the positive real axis, using \(\theta = \text{tan}^{-1}(\frac{y}{x})\)

The correct value of \(\theta\) depends on the quadrant. For 1 - sqrt{5}i, the fourth quadrant complex number:\(\theta = -\text{tan}^{-1}(\text{sqrt}{5})\) By knowing these values, we can convert complex numbers between different forms smoothly.