Problem 52 - Precalculus with Limits

Question

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \((\cos\ 5^{\circ} + i\ \sin\ 5^{\circ})(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})\)

Step-by-Step Solution

Answer

The result of the multiplication is \(\cos\ 25^{\circ} + i\ \sin\ 25^{\circ}\).

Step 1: Identify the trigonometric form

Firstly, you notice that the given complex numbers are in the trigonometric form \(a\cos\theta + b\sin\theta\), where \(a\) and \(b\) are the moduli (lengths) and \(\theta\) is the angle. Here, both \(a\) and \(b\) are equal to 1.

Step 2: Identify the angles

Next, identify the angles from the given complex numbers. In this case, the angles are 5° from \(\cos\ 5^{\circ} + i\ \sin\ 5^{\circ}\) and 20° from \(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ}\).

Step 3: Use De Moivre's theorem to multiply

Now, use De Moivre's theorem to multiply these two complex numbers. The theorem says you just need to add the angles (5° + 20° = 25°) and multiply the lengths. The complex number in trigonometric form from the multiplication will be \(\cos\ 25^{\circ} + i\ \sin\ 25^{\circ}\).

Key Concepts

Complex Numbers

Complex numbers are an essential concept in mathematics, particularly in relation to solving polynomial equations and working with trigonometry and electric circuits. A complex number is composed of a real part and an imaginary part. It is generally expressed in the form of \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Here, \( i \) is the imaginary unit which satisfies the equation \( i^2 = -1 \).
When dealing with trigonometric form, also known as polar form, a complex number can be expressed through angles and moduli. Trigonometric form is often written as \( r(\cos\theta + i\sin\theta) \), where \( r \) is the modulus, or absolute value of the complex number, and \( \theta \) is the argument, representing the angle formed with the positive x-axis.
This form is useful for simplifying multiplication and division of complex numbers, as it leverages the geometric interpretation in the complex plane.

De Moivre's Theorem

De Moivre's Theorem is a fundamental theorem that provides an elegant way to raise complex numbers to integer powers and find roots of complex numbers. It is incredibly powerful when dealing with complex numbers in trigonometric form.
The theorem states that for a complex number expressed as \( (\cos \theta + i \sin \theta) \) and a positive integer \( n \), the formula is:

• \((\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\)

This theorem is also applied when multiplying two complex numbers in trigonometric form. Instead of performing complex algebraic operations, De Moivre's Theorem allows us to simply add their angles and multiply their moduli to find the product of the two complex numbers.
In the exercise provided, by adding the angles 5° and 20°, we directly concluded that the resulting angle is 25°, thanks to De Moivre's Theorem.

Multiplication of Complex Numbers

Multiplication of complex numbers can be quite straightforward when using their trigonometric form. In their standard form, multiplication involves using the distributive property, but this can quickly become cumbersome. By expressing complex numbers as \( \cos\theta + i \sin\theta \), we can use the simplicity of angle addition.
For multiplication:

• Add the angles of the two complex numbers. This gives the angle of the resulting complex number after multiplication.
• Multiply the moduli (absolute values) of the complex numbers. This step simplifies as the modulus is typically \( 1 \) for unit complex numbers in trigonometric form.

Finally, the resultant complex number from multiplication will be in trigonometric form, similar to the original ones, ensuring easy interpretation and visualization.
This method is immensely beneficial, especially for simplifying the multiplication of complex numbers and is widely used in various mathematical and engineering applications.