Problem 65 - 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 \(z=4 i\)

Step-by-Step Solution

Answer

The two roots are \( \sqrt{2} + i\sqrt{2} \) and \( -\sqrt{2} - i\sqrt{2} \).

Step 1: Express in Polar Form

To find the square roots of the complex number \( z = 4i \), first write this number in polar form. The modulus \( r \) is computed as \( r = \sqrt{0^2 + 4^2} = 4 \). The argument \( \theta \) is \( \frac{\pi}{2} \) because \( 4i \) is on the positive imaginary axis. Hence, the polar form is \( 4 \text{cis} \frac{\pi}{2} \), where \( \text{cis} \theta = \cos \theta + i \sin \theta \).

Step 2: Determine the Roots

To find the square roots of \( z \), use the formula for the \( n \)-th roots: if \( w = r \text{cis} \theta \), the \( n \)-th roots are \( w_k = r^{1/n} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \) for \( k = 0, 1, \ldots, n-1 \). Here, \( n = 2 \). So, the roots are \( \sqrt{4} \text{cis} \left( \frac{\frac{\pi}{2}+2k\pi}{2} \right) = 2 \text{cis} \left( \frac{\pi/2+2k\pi}{2} \right) \) for \( k = 0, 1 \).

Step 3: Calculate Each Root in Polar Form

For \( k = 0 \), the angle is \( \frac{\pi/2}{2} = \frac{\pi}{4} \), so one root is \( 2\text{cis} \frac{\pi}{4} \). For \( k = 1 \), the angle is \( \frac{\pi/2+2\pi}{2} = \frac{5\pi}{4} \), so the other root is \( 2\text{cis} \frac{5\pi}{4} \). These are the two square roots in polar form.

Step 4: Convert to Rectangular Form

Convert each polar form back to rectangular form: \( 2\text{cis} \frac{\pi}{4} = 2(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) = 2(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}) = \sqrt{2} + i\sqrt{2} \). For \( 2\text{cis} \frac{5\pi}{4} = 2(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}) = 2(-\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}) = -\sqrt{2} - i\sqrt{2} \). These are the roots in rectangular form.

Key Concepts

Polar form

The polar form of a complex number is a way to express the number using two key components: the modulus and the argument. This form is particularly useful when performing operations like finding roots, as it simplifies calculations involving angles and magnitudes. To write a complex number in polar form, we use the notation:

• \( r \text{cis} \theta \)

where \( r \) is the modulus (or absolute value) of the complex number, and \( \theta \) is the argument (or angle). The term \( \text{cis} \theta \) is shorthand for \( \cos \theta + i \sin \theta \).
In the example with \( z = 4i \), the polar form would be \( 4 \text{cis} \frac{\pi}{2} \). The modulus 4 represents the distance of the complex number from the origin in the complex plane, and \( \frac{\pi}{2} \) is the angle measured counter-clockwise from the positive real axis to the line representing the complex number.

Rectangular form

When expressed in rectangular form, a complex number looks like a usual coordinate pair in the complex plane. It is written in the form of:

• \( a + bi \)

where \( a \) is the real part, and \( bi \) is the imaginary part. Each complex number can be uniquely represented in this form, making it easy to visualize on the complex plane.
For the roots of \( z = 4i \), we calculated their polar form representations first and then converted them to rectangular form. For instance, one root was expressed in polar as \( 2\text{cis} \frac{\pi}{4} \) which, when converted to rectangular form, yields \( \sqrt{2} + i\sqrt{2} \). By using trigonometric identities, we turned the polar components into real and imaginary parts.

Modulus and Argument

The modulus and argument are vital components of the polar form of a complex number. The modulus, denoted by \( r \), measures how far the number is from the origin in the complex plane. Meanwhile, the argument, denoted by \( \theta \), indicates the direction or angle from the positive real axis to the number.

• To compute the modulus, you use the formula: \( r = \sqrt{a^2 + b^2} \)
• The argument is found using: \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)

For\( z = 4i \), the modulus calculated is 4, telling us that 4i is 4 units away from the origin, and the argument \( \frac{\pi}{2} \) signifies it points directly up the imaginary axis.

n-th roots formula

Finding the n-th roots of a complex number becomes much simpler using its polar form. The n-th roots formula allows calculating all possible roots quickly and clearly. The general formula for determining the n-th roots from a complex number \( z = r \text{cis} \theta \) is:

• \[ w_k = r^{1/n} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right),\quad k = 0, 1, \ldots, n-1 \]

This formula helps us find each distinct root by adjusting the angle \( \theta \) with a full circle \( 2\pi \) divided by the number of roots \( n \).
For \( z = 4i \), for which we seek square roots (\( n = 2 \)), the roots are determined as \( 2 \text{cis} \frac{\pi/4} \) and \( 2 \text{cis} \frac{5\pi/4} \). These angles reflect the positions of the roots relative to the complex plane. Using this method systematically, we find all n-th roots, exploiting the symmetrical nature of angles in the polar form.