Problem 58 - Precalculus Mathematics for Calculus

Question

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\). Express your answer in polar form. $$z_{1}=\sqrt{2}\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)$$ $$z_{2}=3 \sqrt{2}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)$$

Step-by-Step Solution

Answer

Product: \( 6 (\cos 135^\circ + i \sin 135^\circ) \), Quotient: \( \frac{1}{3} (\cos 15^\circ + i \sin 15^\circ) \)."

Step 1: Identify the Moduli

The modulus of a complex number in polar form \( z = r (\cos \theta + i \sin \theta) \) is \( r \). Therefore, the modulus for \( z_1 \) is \( \sqrt{2} \) and for \( z_2 \) is \( 3\sqrt{2} \).

Step 2: Determine the Argument of Product

To find the argument of the product \( z_1 z_2 \), add the angles (arguments) of each complex number: \( 75^\circ + 60^\circ = 135^\circ \).

Step 3: Compute the Modulus of the Product

Multiply the moduli of \( z_1 \) and \( z_2 \). The modulus of the product is \( \sqrt{2} \times 3\sqrt{2} = 3 \times 2 = 6 \).

Step 4: Formulate the Product in Polar Form

Combine the modulus and argument of the product to express it in polar form: \( 6 (\cos 135^\circ + i \sin 135^\circ) \).

Step 5: Determine the Argument of the Quotient

For the argument of the quotient \( z_1 / z_2 \), subtract the arguments: \( 75^\circ - 60^\circ = 15^\circ \).

Step 6: Compute the Modulus of the Quotient

Divide the moduli to find the modulus of the quotient: \( \frac{\sqrt{2}}{3\sqrt{2}} = \frac{1}{3} \).

Step 7: Formulate the Quotient in Polar Form

Combine the modulus and argument of the quotient to get \( \frac{1}{3} (\cos 15^\circ + i \sin 15^\circ) \).

Key Concepts

Polar Form

The polar form of a complex number is a way to express the number in terms of its magnitude (modulus) and direction (argument). To transform a complex number from its rectangular form \( a + bi \) into polar form, we use the expression \( z = r(\cos \theta + i \sin \theta) \). Here, \( r \) is the modulus of the complex number, and \( \theta \) is the argument. This form is particularly useful for multiplying and dividing complex numbers, as it simplifies the operations to simple arithmetic with the modulus and the addition or subtraction of angles.

• Magnitude is represented as the modulus \( r \).
• Direction is shown using the argument \( \theta \).
• Expresses complex numbers as \( r(\cos \theta + i \sin \theta) \), also written as \( r \text{cis} \theta \) for short.

Understanding polar form is crucial when performing operations on complex numbers such as finding products or quotients.

Modulus

The modulus of a complex number is essentially its distance from the origin on the complex plane. For a complex number written as \( z = a + bi \), the modulus \( r \) is the square root of the sum of the squares of its real part \( a \) and imaginary part \( b \). In mathematical terms, this is represented as \( r = \sqrt{a^2 + b^2} \).

• The modulus is always a non-negative real number.
• It represents the magnitude of the complex number.
• In polar form, it is the value \( r \) in \( r(\cos \theta + i \sin \theta) \).

When solving problems involving products or quotients of complex numbers, the modulus plays a key role as it simplifies the operation. You multiply the moduli for products and divide them for quotients.

Argument

The argument of a complex number is the angle the complex number makes with the positive real axis on the complex plane. For a complex number \( z = a + bi \), the argument \( \theta \) is found using the inverse tangent function: \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \).

• Angles are often measured in degrees or radians.
• It shows the direction of the complex number in the complex plane.
• In polar form, it is written as \( \theta \) in \( r(\cos \theta + i \sin \theta) \).

When determining the product or quotient of complex numbers, the arguments are simply added or subtracted. This feature makes calculations straightforward, whether you're dealing with multiplication or division.

Product and Quotient of Complex Numbers

Working with complex numbers in polar form simplifies finding their product and quotient. The rules are quite straightforward thanks to the nature of polar coordinates. Let's break it down:

• Product: To find the product of two complex numbers, multiply the moduli and add the arguments: \( r_1 \times r_2 \) and \( \theta_1 + \theta_2 \).
• Quotient: For the quotient, divide the moduli and subtract the arguments: \( \frac{r_1}{r_2} \) and \( \theta_1 - \theta_2 \).
• The operation results are expressed back in polar form, which is \( r(\cos \theta + i \sin \theta) \).

For example, in the given exercise, the product and quotient of \( z_1 \) and \( z_2 \) are formed by using these straightforward rules. Understanding and using these rules makes handling complex numbers efficient and less prone to error.