Problem 47 - Precalculus with Limits

Question

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\left[2 \left(\cos\ \dfrac{\pi}{4} + i\ \sin\ \dfrac{\pi}{4} \right) \right] \left[6 \left(\cos\ \dfrac{\pi}{12} + i\ \sin\ \dfrac{\pi}{12} \right) \right]\)

Step-by-Step Solution

Answer

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

Step 1: Determine the modulus of the product

Calculate the modulus of the resulting complex number which is the product of the moduli of the given complex numbers. In this exercise, the modulus of the first complex number is 2 and of the second complex number is 6. So, the modulus of the product will be \(2 \cdot 6 = 12\).

Step 2: Determine the argument of the product

Calculate the argument of the resulting complex number, which is the sum of the arguments of the given complex numbers. The argument of the first complex number is \(\frac{\pi}{4}\), and the argument of the second complex number is \(\frac{\pi}{12}\). Adding these two arguments results in an argument for the resulting complex number of \(\frac{\pi}{4} + \frac{\pi}{12} = \frac{\pi}{3}\). If the argument is not in the standard range \([- \pi, \pi]\), reduce it to an angle within this interval.

Step 3: Write the final expression in trigonometric form

Write the resulting complex number in trigonometric form using the calculated modulus and argument, which are 12 and \(\frac{\pi}{3}\) respectively. Therefore, the product of the two complex numbers is written as \(12\ (\cos \frac{\pi}{3} + i\ \sin \frac{\pi}{3}) \).

Key Concepts

Understanding Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. These numbers are usually expressed in the form \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the imaginary part, attached to the imaginary unit \( i \), which is defined as \( i^2 = -1 \).

Complex numbers are useful in various fields, including engineering, physics, and mathematics, because they can model two-dimensional quantities such as electrical currents or waves. A complex number can also be represented graphically as a point in a plane, known as the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

By visualizing complex numbers on this plane, operations like addition, subtraction, or multiplication become more intuitive through geometric interpretations. Importantly, combining two or more complex numbers involves operating both their real and imaginary components, which can be efficiently expressed using trigonometric forms.

The Modulus of a Complex Number

The modulus of a complex number is a measure of its "size" or "length" in the complex plane. For a complex number \( z = a + bi \), the modulus is found using the formula:\[ |z| = \sqrt{a^2 + b^2} \]In trigonometric form, when a complex number is represented as \( r(\cos \theta + i \sin \theta) \), the modulus is just \( r \).

• It tells you how far the number is from the origin (0,0) in the complex plane.
• When multiplying two complex numbers, their moduli are also multiplied.

In the case of the exercise, multiplying two complex numbers with moduli 2 and 6 results in a new modulus of \( 12 \). This product reflects the geometric interpretation: when you add the vectors (or rotate and scale them), the length from the origin changes proportionally. Understanding this helps in recognizing patterns in multiplication and predicting the effect on the resulting complex magnitude.

Finding the Argument of a Complex Number

The argument of a complex number is the angle \( \theta \) that the line (or vector) representing the complex number makes with the positive direction of the real axis (x-axis) in the complex plane. It is typically measured in radians and usually falls within the interval \([-\pi, \pi]\).

If a complex number is written in the form \( z = r(\cos \theta + i \sin \theta) \), then \( \theta \) is its argument. The argument can also be calculated from its real and imaginary components using the arctangent function: \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \).

• The main point to remember is that the argument gives the direction angle of the vector in the plane.
• When multiplying two complex numbers, the arguments add up.

In the exercise, adding the arguments \( \frac{\pi}{4} \) and \( \frac{\pi}{12} \) yields an argument for the resulting product of \( \frac{\pi}{3} \). This sum translates geometrically by rotating the vector further in the plane, maintaining the overall direction relative to both components.