Problem 26 - Precalculus : Building Concepts and Connections

Question

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{4}{3}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) \cdot 2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The product of the two given complex numbers in trigonometric form is \( \frac{8}{3}\left(\cos \frac{7\pi}{12}+i \sin \frac{7\pi}{12}\right) \).

Step 1: Simplification

First simplify the problem by multiplying the magnitudes and adding the arguments. The magnitude of the product is the product of the magnitudes, which is \( \frac{4}{3} \cdot 2 = \frac{8}{3} \). The argument of the product is the sum of the arguments, which is \( \frac{\pi}{3} + \frac{\pi}{4} = \frac{7\pi}{12} \). Thus, the product can be expressed as \( \frac{8}{3}(\cos \frac{7\pi}{12}+i \sin \frac{7\pi}{12}) \).

Step 2: Converting into Trigonometric Form

Now, substitute these into the trigonometric form. We get \( \frac{8}{3}\left(\cos \frac{7\pi}{12}+i \sin \frac{7\pi}{12}\right) \).

Step 3: Final Answer

The expression is now already in the final form. Hence, the multiplication of the complex numbers in trigonometric form results in \( \frac{8}{3}\left(\cos \frac{7\pi}{12}+i \sin \frac{7\pi}{12}\right) \).

Key Concepts

Multiplication of Complex Numbers

Multiplying complex numbers can be easier when they are expressed in trigonometric form, especially when you deal with the angles and moduli separately. When a complex number is given in the form \( r (\cos \theta + i \sin \theta) \), multiplication follows this general rule:

• Multiply the magnitudes: This step involves multiplying the real number parts (the magnitudes) together. For example, if you have magnitudes of \( \frac{4}{3} \) and \( 2 \), their product is \( \frac{8}{3} \).
• Add the angles: For the angles \( \theta \), you simply add them together. Using our example, adding \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) results in \( \frac{7\pi}{12} \).

This avoids dealing with complex parts directly and keeps everything organized. Multiplication in trigonometric form can simplify computation, especially when angles and magnitudes are not straightforward combinations.

Trigonometric Identities

Trigonometric identities are very useful in complex number calculations, especially when angles are involved. These identities help simplify expressions when working with complex numbers in trigonometric form. Some important identities to remember include:

• Angle Addition Identity: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \) and \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \).
• Conversion of angles: Understanding how to convert angles between degrees and radians can be helpful in ensuring your calculations are consistent.

These can be particularly helpful when simplifying the arguments of complex numbers during multiplication, ensuring that your resulting expressions are as simplified as possible. Moreover, recognizing these identities can often reveal more opportunities for further simplification or manipulation!

Argument of a Complex Number

The argument of a complex number refers to the angle formed with the positive real axis in the complex plane. In trigonometric form, it is usually expressed with the angle \( \theta \) in \( r (\cos \theta + i \sin \theta) \). Understanding the argument is vital because:

• Addition and Subtraction: It helps in determining how to add or subtract angles when multiplying or dividing complex numbers in trigonometric forms. For example, when multiplying two complex numbers, you add their arguments as illustrated in \( \frac{7\pi}{12} \).
• Determining Direction: The argument is also a visual and directional representation of where the complex number lies on the complex plane.

Mastering the concept of an argument allows you to smoothly perform operations and transitions between different forms, such as moving from algebraic to trigonometric forms.