Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\left[3\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\right]\left[9\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)\right]$$

Step-by-Step Solution

Answer

The product of the given complex numbers in trigonometric form is \( 27(\cos(2\pi) + i \sin (2\pi)) \).

Step 1: Identify the Modulus and Argument

Firstly, to perform the operation, one must identify the modulus (r) and arguments (θ) of the complex numbers. For the first complex number we have \( r_1 = 3 \) and \( θ_1 = \frac{\pi}{2} \) . For the second complex number we have \( r_2 = 9 \) and \( θ_2 = \frac{3\pi}{2} \).

Step 2: Apply the Properties of Modulus and Argument

Next, apply the properties of modulus and argument to the identified numbers. The modulus of the product of two complex numbers is the product of their moduli and the argument of the product is the sum of their arguments. After applying these properties we get our new modulus: \( r = r_1 * r_2 = 3 * 9 = 27 \) and new argument: \( θ = θ_1 + θ_2 = \frac{\pi}{2} + \frac{3\pi}{2} = 2\pi \).

Step 3: Formulate the Result in Trigonometric Form

The final step is to re-write the result in trigonometric form. This gives us: \( 27(\cos(2\pi) + i \sin (2\pi)) \). This is the product of the given complex numbers.

Key Concepts

Understanding Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. In complex numbers, \( i \) is the imaginary unit, defined by the property \( i^2 = -1 \). This opens up a whole new dimension to handle numbers that aren’t on the standard number line.
Complex numbers are also visualized on the complex plane, where the x-axis represents real numbers, and the y-axis represents imaginary numbers.

• The point \( (a, b) \) on this plane represents the complex number \( a + bi \).
• Every complex number can have a corresponding magnitude and direction from the origin, giving rise to their trigonometric representation.

This trigonometric form is often useful for geometric interpretations and operations like multiplication.

Modulus and Argument: The Heart of Trigonometric Form

The modulus and argument are key concepts in understanding the trigonometric form of complex numbers.

• Modulus is the distance of the complex number from the origin on the complex plane. It is found using the Pythagorean theorem: \( |a + bi| = \sqrt{a^2 + b^2} \).
• Argument is the angle measured from the positive x-axis to the line segment joining the origin and the point \( (a, b) \). It is denoted as \( \theta \), and typically calculated using the \( \tan^{-1}(\frac{b}{a}) \).

A complex number in trigonometric form is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus, and \( \theta \) is the argument. This form helps to perform operations such as multiplication and division more easily.

Operations on Complex Numbers

Operations on complex numbers, like addition, subtraction, multiplication, and division, are enhanced by using their trigonometric form.

For multiplication, as in the provided exercise, the modulus of the product of two complex numbers is simply the product of their individual moduli. Similarly, the argument of the product is the sum of their individual arguments:

• Given two complex numbers \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), their product \( z_1z_2 \) is
• \( r_1r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \).

This operation is much simpler than expanding and reducing terms manually in rectangular form.
The trigonometric form not only simplifies these calculations but also provides a clear geometric interpretation, thus enhancing one's understanding of complex number operations.