Problem 80 - Algebra and Trigonometry

Question

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ \frac{(-1+i \sqrt{3})(2-2 i \sqrt{3})}{4 \sqrt{3}-4 i} $$

Step-by-Step Solution

Answer

Polar form: \(e^{iπ/6}\), Rectangular form: \(\frac{\sqrt{3}}{2} + \frac{i}{2}\)

Step 1: Convert to Polar Form

Firstly, write the complex numbers in polar form using the formula \(r = \sqrt{x^2 + y^2}\) and \(θ = arctan(\frac{y}{x})\).The numbers can be written as:\((-1+i \sqrt{3}) = 2e^{i2θ}\) where \(r=2\) and \(θ=\frac{π}{3}\),\((2-2 i \sqrt{3}) = 4e^{i4θ}\) where \(r=4\) and \(θ=-\frac{π}{3}\),\((4 \sqrt{3}-4 i) = 8e^{-iθ}\) where \(r=8\) and \(θ=\frac{\pi}{2}\).Use these polar forms for the operation.

Step 2: Perform the Operation

Once the conversion is done, we can perform the operation. The division of polar forms is performed by subtracting angles and dividing magnitudes. The operation leads to\(\frac{(2e^{i2π/3})*(4e^{-i4π/3})}{8e^{-iπ/2}}\) which simplifies to\(e^{iπ/6}\), in polar form.

Step 3: Convert to Rectangular Form

Convert it back to the rectangular form using Euler's formula, which is \(x + yi = r(cosθ + isinθ)\). The rectangular form of \(e^{iπ/6}\) is \(\frac{\sqrt{3}}{2} + \frac{i}{2}\).

Key Concepts

Polar Form

Polar form is a way to express complex numbers that highlights their geometric properties.
Instead of using coordinates on the real and imaginary axes, polar form uses a radius and an angle.
Any complex number can be converted into polar form using the formula:

• The radius or modulus: \( r = \sqrt{x^2 + y^2} \)
• The angle or argument: \( \theta = \text{arctan}\left(\frac{y}{x}\right) \)

Combining these, a complex number \( a + bi \) becomes \( re^{i\theta} \) in polar form. In the given problem, each complex number is represented in this form before performing any mathematical operations. This is because multiplication and division in polar form are easier to handle.

Rectangular Form

Rectangular form is the standard way of representing complex numbers using Cartesian coordinates.
A complex number in rectangular form can be written as \( x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.
Converting from polar to rectangular form involves using the relationships:

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

This makes it easy to switch between forms using Euler's formula, especially once calculations are made in polar form.
In the exercise, after deriving the result in polar form, converting back to rectangular form allowed the solution to be expressed as \( \frac{\sqrt{3}}{2} + \frac{i}{2} \). Using both forms as needed leverages their respective strengths.

Euler's Formula

Euler's formula is a powerful tool in complex numbers, linking together polar and rectangular forms.
It states: \[ e^{i\theta} = \cos \theta + i\sin \theta \] Euler's formula beautifully unites the geometry of complex numbers with their algebraic properties.
When you express a complex number in polar form as \( re^{i\theta} \), it can be converted readily to rectangular form \( r(\cos \theta + i\sin \theta) \) using Euler's insight.
This conversion was crucial in the provided solution, allowing the conversion of the polar result \( e^{i\pi/6} \) back to \( \frac{\sqrt{3}}{2} + \frac{i}{2} \).Understanding Euler's formula aids in seamlessly switching back and forth between forms.

Complex Division

Division of complex numbers can be quite facilitated by working in polar form.
In polar form, division becomes straightforward: divide the magnitudes and subtract the angles.
This property makes polar form very attractive for complex divisions.In the original exercise, we took advantage of this by converting three complex numbers, \((-1+i \sqrt{3}), (2-2 i \sqrt{3}), \text{and} (4 \sqrt{3}-4 i)\), into polar form. Each was expressed with its own radius and angle, then combined using the rule of subtracting the angles and dividing the magnitudes:

• \( \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \)

This transformation allowed us to compute the division accurately, yielding a result that was then reverted to rectangular form using the knowledge of polar division. By understanding the geometric interpretation behind this process, students can grasp why polar form simplifies these operations.