Problem 68 - College Trigonometry

Question

Find the indicated complex roots. Express your answers in polar form and then convert them into rectangular form. the two square roots of \(\frac{5}{2}-\frac{5 \sqrt{3}}{2} i\)

Step-by-Step Solution

Answer

The square roots in rectangular form are \( \frac{\sqrt{15}}{2} - \frac{\sqrt{5}}{2}i \) and \( -\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2}i \).

Step 1: Convert to Polar Form

To find the polar form of the complex number \( \frac{5}{2} - \frac{5 \sqrt{3}}{2} i \), calculate its magnitude and argument. The magnitude \( r \) is given by \( r = \sqrt{ \left( \frac{5}{2} \right)^2 + \left( -\frac{5 \sqrt{3}}{2} \right)^2 } \), which simplifies to \( r = 5 \). The argument \( \theta \) is \( \tan^{-1}\left( \frac{-\frac{5 \sqrt{3}}{2}}{\frac{5}{2}} \right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \). Thus, the polar form is \( 5 \text{cis}(-\frac{\pi}{3}) \).

Step 2: Find the Square Roots in Polar Form

The square roots of a complex number in polar form \( r \text{cis}(\theta) \) are given by \( \sqrt{r} \text{cis}\left( \frac{\theta + 2k\pi}{n} \right) \) for \( n = 2 \). Here \( \sqrt{5} \text{cis}\left(-\frac{\pi}{6} + k\pi \right) \) for \( k = 0, 1 \) since \( n = 2 \). This gives us two roots in polar form: \( \sqrt{5} \text{cis}(-\frac{\pi}{6}) \) and \( \sqrt{5} \text{cis}(\frac{5\pi}{6}) \).

Step 3: Convert Roots to Rectangular Form

Convert each root from polar to rectangular form using \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \). For \( \sqrt{5} \text{cis}(-\frac{\pi}{6}) \), \( a = \sqrt{5} \cos(-\frac{\pi}{6}) = \sqrt{5} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{15}}{2} \) and \( b = \sqrt{5} \sin(-\frac{\pi}{6}) = \sqrt{5} \times -\frac{1}{2} = -\frac{\sqrt{5}}{2} \). So, the rectangular form is \( \frac{\sqrt{15}}{2} - \frac{\sqrt{5}}{2}i \). For \( \sqrt{5} \text{cis}(\frac{5\pi}{6}) \), \( a = \sqrt{5} \cos(\frac{5\pi}{6}) = \sqrt{5} \times -\frac{\sqrt{3}}{2} = -\frac{\sqrt{15}}{2} \) and \( b = \sqrt{5} \sin(\frac{5\pi}{6}) = \sqrt{5} \times \frac{1}{2} = \frac{\sqrt{5}}{2} \). Thus, the rectangular form is \( -\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2}i \).

Key Concepts

Polar Form

The polar form of a complex number is a way of expressing the number using a magnitude and an angle. Normally, a complex number is written as \( a + bi \), known as the rectangular form. However, in terms of polar form, a complex number can be expressed as \( r \text{cis} \theta \), where \( r \) represents the magnitude (or modulus) and \( \theta \) is the argument (or angle).

The term "cis" is a shorthand for \( \cos \theta + i \sin \theta \).

This form is very useful when multiplying or finding roots of complex numbers, as it simplifies the operations. To convert from rectangular to polar form, we use the formulas for magnitude and argument, which depend on the components \( a \) (the real part) and \( b \) (the imaginary part) of the complex number.

Rectangular Form

Rectangular form is the standard way to express complex numbers as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

This presentation is similar to a point in a two-dimensional Cartesian coordinate system, where \( a \) is the x-coordinate and \( b \) is the y-coordinate. This representation is very intuitive when dealing with addition or subtraction of complex numbers.

However, it can be cumbersome with certain operations like multiplication and division, which is why polar form is sometimes preferred. Despite this, converting from polar back to rectangular form is simple with the equations:

• \( a = r \cos(\theta) \)
• \( b = r \sin(\theta) \)

This lets you reverse the process and retrieve the real and imaginary components from the polar form.

Magnitude and Argument

Key to understanding and working with complex numbers is the magnitude and argument. The magnitude of a complex number \( a + bi \) is determined by \( r = \sqrt{a^2 + b^2} \).

This provides the length of the vector representation of the complex number. Meanwhile, the argument \( \theta \) is the angle made with the positive real axis, given by \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

• The magnitude tells us about the size or distance of the complex number from the origin.
• The argument or phase indicates the direction.

These measures are fundamental in converting to polar form and are crucial for operations such as finding roots or products, as these calculations are simply adjustments in magnitude and angle.

Converting Forms

Converting between rectangular and polar forms involves using the magnitude and argument, as these act as a bridge between the two representations.

To convert a complex number from rectangular \( a + bi \) to polar form \( r \text{cis} \theta \), use:

• \( r = \sqrt{a^2 + b^2} \)
• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

When going from polar back to rectangular, apply:

• \( a = r \cos(\theta) \)
• \( b = r \sin(\theta) \)

These conversions are handy for solving complex problems, like finding roots of a complex number, as they often involve breaking tasks down into simpler, more manageable parts facilitated by polar representation.