Problem 27 - Precalculus : Building Concepts and Connections

Question

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{5\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)}{2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)}$$

Step-by-Step Solution

Answer

\(\frac{5}{2} (\cos \frac{\pi}{12} + i \sin \frac{\pi}{12})\)

Step 1: Identify the Magnitudes and Angles

Identify the magnitudes (or moduli) and the angles (or arguments) of the complex numbers. In this case, for the numerator \(5 (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\), the magnitude is 5 and the angle is \(\frac{\pi}{4}\). For the denominator \(2 (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\), the magnitude is 2 and the angle is \(\frac{\pi}{6}\).

Step 2: Divide the Magnitudes and Subtract the Angles

Follow the rules for dividing complex numbers in trigonometric form. Divide the magnitudes and subtract the angles. The magnitude of the result is \(\frac{5}{2}\) and the angle is \(\frac{\pi}{4} - \frac{\pi}{6}\).

Step 3: Simplify the Result

Simplify the resulting complex number. The magnitude of \(\frac{5}{2}\) remains as it is, and the angle simplifies to \(\frac{\pi}{12}\).

Step 4: Write the Result in Trigonometric Form

Write the result in trigonometric form. This gives \(\frac{5}{2} (\cos \frac{\pi}{12} + i \sin \frac{\pi}{12})\) as the final result.

Key Concepts

Exploring Complex Numbers

Complex numbers are a fascinating and crucial concept in mathematics, extending the traditional idea of number to include both real and imaginary components. Imagine numbers shaped as ordered pairs like (3, 5), where 3 is real and 5 is imaginary.
This pair can be denoted as 3 + 5i, where 'i' is the imaginary unit, defined as the square root of -1.

• The real part of a complex number is a regular number that we are familiar with, like 3 in our example.
• The imaginary part is always multiplied by the imaginary unit 'i'.

Understanding these components helps us to perform operations such as addition, subtraction, multiplication, and division on complex numbers. The trigonometric form is another way to express these numbers, focusing primarily on their magnitudes and directions, which makes them very applicable across fields like engineering and physics.

Understanding Magnitudes and Angles

Magnitudes and angles are vital when dealing with complex numbers in trigonometric form. This form is ideal for representing complex numbers on a plane by focusing on their size and direction, rather than just their position.

• The magnitude of a complex number is its distance from the origin in the complex plane and can be found using the formula \(|z| = \sqrt{a^2 + b^2}\), where the complex number is represented as \(z = a + bi\).
• The angle, also called the argument, tells us the angle the line from the origin to the point makes with the positive real axis. It's usually denoted by \(\theta\).

When multiplying or dividing complex numbers in trigonometric form, these elements attend mainly to the rules of manipulating magnitudes and angles. Multiplication involves multiplying the magnitudes and adding the angles. For division, we divide the magnitudes and subtract the angles. This systematic approach simplifies complex operations.

The Role of Trigonometric Identities

Trigonometric identities simplify expressions and calculations involving trigonometric functions by establishing relationships between different angles and sides of a triangle.
This knowledge extends to complex numbers in trigonometric form.

• They allow us to write complex numbers efficiently as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the angle.
• Identities such as \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\) help in manipulating angles during division or multiplication of complex numbers.

Learning these identities is incredibly useful when simplifying the results of complex number operations. They provide robust tools for transforming expressions into forms that are easier to interpret and work with. In our example, using these identities allows us to simplify the angle to lie between standard intervals, ensuring trigonometric forms remain clear and standardized.