Problem 66 - Algebra and Trigonometry

Question

In Exercises \(65-68\), find all the complex roots. Write roots in polar form with \(\theta\) in degrees. The complex square roots of \(25\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)\)

Step-by-Step Solution

Answer

The square roots of \(25(\cos 210^{\circ}+i \sin 210^{\circ})\) are \(5(\cos 105^{\circ}+i \sin 105^{\circ})\) and \(5(\cos 285^{\circ}+i \sin 285^{\circ})\)

Step 1: Express the complex number in polar form

The complex number is already in polar form: \(25(\cos 210^{\circ}+i \sin 210^{\circ})\). In this case, \(r=25\) is the magnitude and \(\theta=210^{\circ}\) is the argument.

Step 2: Apply the formula for square root of a complex number

To find square roots of a complex number \(z = r(\cos \theta + i \sin \theta)\), we apply the formula \(\sqrt{r}(\cos \frac{\theta}{2} + i \sin \frac{\theta}{2})\). In this case, the square root of the magnitude \(r=25\) is \(\sqrt{25}=5\), and dividing the angle by 2 gives \(\frac{210}{2}=105^{\circ}\). Therefore, one root is \(5(\cos 105^{\circ}+i \sin 105^{\circ})\).

Step 3: Find the other root

The other root can be obtained by adding \(180^{\circ}\) to the argument. This is because squaring the root will still yield the original complex number due to the periodicity of the cosine and sine functions (being \(360^{\circ}\)), hence \(5(\cos (105^{\circ} + 180^{\circ})+i \sin (105^{\circ} + 180^{\circ})) = 5(\cos 285^{\circ}+i \sin 285^{\circ})\) is the other root.

Key Concepts

Polar Form

The polar form of a complex number is a very handy way of representing complex numbers. Unlike the standard form, which uses a real part and an imaginary part, the polar form consists of a magnitude, also known as "modulus," and an angle, known as "argument." This form is particularly useful in problems involving multiplication, division, and finding roots.

A complex number in polar form is written as \( r(\cos \theta + i \sin \theta) \). Here:

• \( r \) is the magnitude or absolute value, which tells us how far the number is from the origin on the complex plane.
• \( \theta \) is the argument, and it indicates the direction of the number.

For example, in the given exercise, the complex number is \( 25(\cos 210^{\circ} + i \sin 210^{\circ}) \). This means the number's magnitude is 25, and it makes an angle of \( 210^{\circ} \) with the positive real axis.

By expressing complex numbers in polar form, we simplify the process of finding their roots, making calculations more intuitive and manageable.

Complex Roots

Finding complex roots is exciting because it often uncovers multiple solutions due to the nature of complex numbers. When you find the root of a complex number, you are essentially looking for other numbers that can be combined to recreate the original number when multiplied together.

For a complex number in the form \( z = r(\cos \theta + i \sin \theta) \), the \( n^{th} \) root is given by:

• \( \sqrt[n]{r}(\cos \frac{\theta + 360^{\circ}k}{n} + i \sin \frac{\theta + 360^{\circ}k}{n}) \), where \( k = 0, 1, 2, \ldots, n-1 \).

In the exercise, we found the square roots of \( 25(\cos 210^{\circ} + i \sin 210^{\circ}) \). The calculation involves dividing the angle by 2 and taking the square root of the magnitude, yielding two roots:

• First Root: \( 5(\cos 105^{\circ} + i \sin 105^{\circ}) \)
• Second Root: \( 5(\cos 285^{\circ} + i \sin 285^{\circ}) \)

Each root corresponds to a different direction in the complex plane, but both give the original number when squared. Complex roots thus unfold the elegant symmetry and periodicity of the complex number system.

Trigonometric Form of Complex Numbers

The trigonometric form is essentially the same as the polar form, as it uses sine and cosine to represent the complex number's angle. However, focusing on its trigonometric components can make certain mathematical operations, like finding powers and roots, significantly easier.

This approach uses Euler's formula, which states that \( e^{i\theta} = \cos \theta + i \sin \theta \). It connects exponential functions with trigonometric functions, paving the way for an elegant form of complex numbers.

• This form is particularly useful for visualizing and calculating the rotation and reflection operations on the complex plane.
• It helps simplify the multiplication and division of complex numbers, by transforming angle addition and subtraction into simple trigonometric operations.

In the exercise, the complex number is already given in trigonometric form as \( 25(\cos 210^{\circ} + i \sin 210^{\circ}) \). When finding the roots, the magnitude changes to \( \sqrt{25} = 5 \), and the angle is halved to reflect the square roots, accompanied by an extra root found by adding \( 180^{\circ} \), showing how trigonometric form facilitates the extraction of complex roots.