Problem 53 - Algebra and Trigonometry

Question

\(49-56\) me product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)} \\\ {z_{2}=2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

Product: 8(cos 150° + i sin 150°), Quotient: 2(cos 90° + i sin 90°).

Step 1: Convert to Polar Form

Both complex numbers are already given in polar form: \[ z_1 = 4 \left( \cos 120^\circ + i \sin 120^\circ \right) \] and \[ z_2 = 2 \left( \cos 30^\circ + i \sin 30^\circ \right) \]. The modulus of \(z_1\) is 4, and its argument is 120°.The modulus of \(z_2\) is 2, and its argument is 30°.

Step 2: Calculate the Product

To find the product of two complex numbers in polar form, multiply their moduli and add their arguments. \[ |z_1 \, z_2| = 4 \times 2 = 8 \]\[ \arg(z_1 \, z_2) = 120^\circ + 30^\circ = 150^\circ \]Thus, the product is \[ z_1 \, z_2 = 8 \left( \cos 150^\circ + i \sin 150^\circ \right) \].

Step 3: Calculate the Quotient

To find the quotient of two complex numbers in polar form, divide their moduli and subtract their arguments. \[ \left|\frac{z_1}{z_2}\right| = \frac{4}{2} = 2 \]\[ \arg\left(\frac{z_1}{z_2}\right) = 120^\circ - 30^\circ = 90^\circ \]Thus, the quotient is \[ \frac{z_1}{z_2} = 2 \left( \cos 90^\circ + i \sin 90^\circ \right) \].

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers, using a combination of their modulus and argument. It is a very useful format, especially when working with complex number multiplication and division.

A complex number in polar form is written as \[ z = r (\cos \theta + i \sin \theta) \]where:

• \(r\) is the modulus or the magnitude of the complex number.
• \(\theta\) is the argument or the angle it forms with the real axis.

By converting a complex number to polar form, calculations such as multiplication and division become much simpler. This is because you can operate directly on the modulus and argument for these calculations.

Modulus

The modulus of a complex number, denoted as \(r\), is its distance from the origin in the complex plane.

To find the modulus of a complex number in the form \(z = a + bi\), you use the formula:\[|z| = \sqrt{a^2 + b^2}\]However, when a complex number is in polar form, the modulus is immediately evident. For example, in \(z_1 = 4 (\cos 120^\circ + i \sin 120^\circ)\), the modulus is 4.

The modulus is always a non-negative real number, representing the "size" of the complex number irrespective of its direction.

Argument

The argument of a complex number, denoted as \(\theta\), is the angle that the line representing the complex number makes with the positive real axis.

Typically, the argument is measured in degrees or radians. For instance, in the polar form \(z_2 = 2 (\cos 30^\circ + i \sin 30^\circ)\), the argument is 30°. It tells you the direction of the complex number in the complex plane.

In operations like complex multiplication and division, the arguments are particularly important as they determine how to rotate the number in the complex plane.

Complex Multiplication

Complex multiplication is remarkably straightforward in polar form. When you multiply two complex numbers:

• Multiply their moduli: \(|z_1 z_2| = |z_1| \times |z_2|\)
• Add their arguments: \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\)

For example, if \(z_1 = 4 (\cos 120^\circ + i \sin 120^\circ)\) and \(z_2 = 2 (\cos 30^\circ + i \sin 30^\circ)\), their product in polar form is:\[ z_1 z_2 = 8 (\cos 150^\circ + i \sin 150^\circ) \]This makes the multiplication process very simple and intuitive.

Complex Division

Complex division is also simplified in polar form. When dividing one complex number by another:

• Divide their moduli: \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)
• Subtract their arguments: \(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\)

Consider the case where \(z_1 = 4 (\cos 120^\circ + i \sin 120^\circ)\) and \(z_2 = 2 (\cos 30^\circ + i \sin 30^\circ)\), the quotient is:\[\frac{z_1}{z_2} = 2 (\cos 90^\circ + i \sin 90^\circ)\]This method makes complex division manageable and intuitive, emphasizing mostly on the modulus and angle operations.