Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\left[4\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)\right]\left[5\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right)\right]$$

Step-by-Step Solution

Answer

The product of the given complex numbers in trigonometric form is \(20(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})\).

Step 1: Identify the magnitude and angle of each complex number

The given complex numbers are \(4(\cos \frac{\pi}{12} + i \sin \frac{\pi}{12})\) and \(5(\cos \frac{7\pi}{12} + i \sin \frac{7\pi}{12})\). Here, the magnitudes are 4 and 5 respectively and the angles, \(\theta\), are \(\frac{\pi}{12}\) and \(\frac{7\pi}{12}\) respectively.

Step 2: Perform multiplication of the magnitudes and addition of the angles

The magnitude of the product is obtained by multiplying the magnitudes of the given complex numbers. So, the magnitude of the product is \(4 \times 5 = 20\). The angle of the product is obtained by adding the angles of the given complex numbers. In this case, the angle of the product is \(\frac{\pi}{12} + \frac{7\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3}\)

Step 3: Write the result in the standard form

The product of the given complex numbers is then given by the magnitude times the cosine of the angle plus \(i\) times the sine of the angle. So, the result is \(20(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})\).

Key Concepts

Multiplication of Complex Numbers in Trigonometric Form

Understanding the multiplication of complex numbers can initially seem daunting, but by using the trigonometric form, it can be simplified considerably. When two complex numbers in trigonometric form are multiplied, the magnitudes of these numbers are multiplied together while the angles are added. A general complex number in trigonometric form is represented as \( r(\text{cos} \theta + i \text{sin} \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) the angle (or argument) with the positive x-axis.

Let's break down this process:

• To multiply two complex numbers, multiply their magnitudes to get the magnitude of the result.
• Add their angles to get the angle of the result.
• Express the new complex number using the new magnitude and angle in the trigonometric form.

For an illustrative example, if we have two complex numbers \( 4(\text{cos} \frac{\pi}{12} + i \text{sin} \frac{\pi}{12}) \) and \( 5(\text{cos} \frac{7\pi}{12} + i \text{sin} \frac{7\pi}{12}) \), multiplying their magnitudes gives us \( 20 \) and adding their angles gives us \( \frac{2\pi}{3} \). Therefore, the product is \( 20(\text{cos} \frac{2\pi}{3} + i \text{sin} \frac{2\pi}{3}) \).

Trigonometric Representation of Complex Numbers

The trigonometric representation of complex numbers is a powerful tool that simplifies many operations, including multiplication and division. A complex number \( z \) can be written in the form \( z = r(\text{cos} \theta + i \text{sin} \theta) \), where \( r \) is the magnitude of \( z \), and \( \theta \) is the angle formed with the positive direction of the real axis (usually measured in radians). This representation ties directly into the geometry of the complex plane.

Each complex number corresponds to a point in the plane, with the magnitude representing the distance from the origin and the angle specifying the direction. A key advantage of the trigonometric form lies in its ability to translate the geometric interpretation of complex numbers directly into algebraic operations, as seen with multiplication and division where magnitudes and angles have clearly-defined and intuitive rules for combination.

Magnitude and Angle of Complex Numbers

The magnitude (or modulus) and angle (or argument) are fundamental characteristics of any complex number when expressed in polar or trigonometric form. The magnitude of a complex number, represented by \( r \), is the distance from the origin to the point in the complex plane, found using the Pythagorean theorem as \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary components, respectively.

The angle, typically denoted as \( \theta \), determines the direction of the number from the origin, measured counterclockwise from the positive x-axis to the line connecting the origin to the point. You can find the angle using the inverse tangent function, but you must consider the sign of the real and imaginary parts to determine the correct quadrant. In short:

• The magnitude quantifies 'how far' the complex number is from the origin.
• The angle indicates 'in which direction' (from the positive real axis) the number is located.
• Together, magnitude and angle give a complete description of the position of a complex number in the plane.

These concepts are especially important when performing multiplications or divisions, as they direct us to the correct geometric interpretation.