Problem 62 - Precalculus

Question

Explain the mistake that is made. Let \(z_{1}=6\left(\cos 65^{\circ}+i \sin 65^{\circ}\right)\) and \(z_{2}=3\left(\cos 125^{\circ}+i \sin 125^{\circ}\right) .\) Find \(z_{1} z_{2}\) Solution: Write the product. \(z_{1} z_{2}=6\left(\cos 65^{\circ}+i \sin 65^{\circ}\right) \cdot 3\left(\cos 125^{\circ}+i \sin 125^{\circ}\right)\). Multiply the magnitudes. $$ z_{1} z_{2}=18\left(\cos 65^{\circ}+i \sin 65^{\circ}\right)\left(\cos 125^{\circ}+i \sin 125^{\circ}\right) $$ Multiply the cosine terms and sine terms (add the argumcnts). $$ z_{1} z_{2}=18\left[\cos \left(65^{\circ}+125^{\circ}\right)+i^{2} \sin \left(65^{\circ}+125^{\circ}\right)\right] $$ Simplify \(\left(i^{2}=-1\right)\) $$ z_{1} z_{2}=18\left(\cos 190^{\circ}-\sin 190^{\circ}\right) $$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The error was incorrectly simplifying the sine term; it should be plus \( i \sin(190^{\circ}) \), not minus.

Step 1: Recall the Representation of Complex Numbers in Polar Form

The complex numbers \( z_1 = 6(\cos 65^{\circ} + i \sin 65^{\circ}) \) and \( z_2 = 3(\cos 125^{\circ} + i \sin 125^{\circ}) \) are given in trigonometric form, also known as polar form. In this form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument of the complex number.

Step 2: Use the Multiplication Rule for Complex Numbers in Polar Form

To find the product \( z_1 z_2 \), we multiply the magnitudes and add the arguments. Thus, \[ z_1 z_2 = |z_1||z_2|(\cos(\arg(z_1) + \arg(z_2)) + i \sin(\arg(z_1) + \arg(z_2))) \].Here, \( |z_1| = 6 \) and \( |z_2| = 3 \), so their product is \( 6 \times 3 = 18 \). The arguments are added: \( 65^{\circ} + 125^{\circ} = 190^{\circ} \).

Step 3: Correct the Formula Application

The mistake occurs when calculating \( z_1 z_2 = 18(\cos(190^{\circ}) + i \sin(190^{\circ})) \). The correct formula for polar multiplication keeps the \( i \) in front of the sine term. Thus, the multiplication does not alter the signs incorrectly:The correct expression should be: \[ z_1 z_2 = 18(\cos 190^{\circ} + i \sin 190^{\circ}) \].

Step 4: Recognize the Error in Misapplying \( i^2 \)

The step that incorrectly simplifies \( i^2 \sin(190^{\circ}) \) to \(-\sin(190^{\circ})\) is wrong because, during multiplication, properties of \( i \) should not change the formula structure: the sine term should remain positive in a standard polar form.

Key Concepts

Polar Form

The polar form of a complex number presents it as a product of its magnitude and a trigonometric expression. A complex number like \(z = r(\cos \theta + i \sin \theta)\) consists of two main components: the modulus \(r\), which is the distance from the origin in the complex plane, and the angle \(\theta\), known as the argument, which is the angle formed with the positive real axis.
The polar form is particularly advantageous when performing operations such as multiplication and division. It simplifies processes that involve complex numbers by converting them into easier numerical calculations. This form highlights the cyclic nature of complex numbers due to their periodic dependence on sine and cosine functions.
When expressing complex numbers in polar form, it’s crucial to maintain the integrity of trigonometric identities. This means respecting the structure \(\cos \theta + i \sin \theta\), where \(i\) is preserved to reflect the imaginary component.

Multiplication of Complex Numbers

In polar form, multiplying complex numbers is straightforward and intuitive. The process relies on multiplying their magnitudes and summing their 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 \(z_1 z_2\) is determined by:

• Multiplying the moduli: \(|z_1||z_2| = r_1 \times r_2\).
• Adding the arguments: \(\theta_1 + \theta_2\).

Overall, this results in the product expressed as:
\[z_1 z_2 = |z_1||z_2|(\cos(\arg(z_1) + \arg(z_2)) + i \sin(\arg(z_1) + \arg(z_2)))\]
This formula highlights that the magnitude of the product is the product of the magnitudes, and the direction is the combined angles. This method retains all properties of complex numbers, including the imaginary component, which is denoted by \(i\), ensuring results remain accurate.

Trigonometric Form

The trigonometric form of a complex number is synonymous with its polar form. It leverages fundamental trigonometric functions—cosine and sine—to represent the position of a complex number on the complex plane. This form \(z = r(\cos \theta + i \sin \theta)\) clearly shows how complex numbers relate to angles and radii.
Understanding trigonometric form aids in visualizing complex numbers' positions and interactions. It’s particularly useful when performing geometric transformations, as trigonometric identities facilitate the elegant handling of rotations and scalings.
In calculations, the trigonometric form is powerful and elegant, particularly for multiplication and division. Its structure incorporates sine and cosine in a way that reveals the underlying rotational and stretching characteristics of complex number operations, like multiplication, making it indispensable in fields that rely on complex numbers, such as engineering and physics.