Problem 61 - Algebra and Trigonometry

Question

\(57-64=\) Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=5+5 i, \quad z_{2}=4 $$

Step-by-Step Solution

Answer

\(z_1 = 5\sqrt{2} e^{i\frac{\pi}{4}}\), \(z_2 = 4\); Product: \(20\sqrt{2} e^{i\frac{\pi}{4}}\); Quotients: \(\frac{5\sqrt{2}}{4} e^{i\frac{\pi}{4}}\), \(\frac{1}{5\sqrt{2}} e^{-i\frac{\pi}{4}}\).

Step 1: Convert Complex Numbers to Polar Form

First, we convert the given complex numbers \(z_1 = 5 + 5i\) into polar form. The modulus \( r_1 \) is calculated as \( \sqrt{(5)^2 + (5)^2} = \sqrt{50} = 5\sqrt{2} \). The argument \( \theta_1 \) is \( \tan^{-1}(\frac{5}{5}) = \frac{\pi}{4} \). So, the polar form of \( z_1 \) is \( 5\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) \). For \( z_2 = 4 \), which is a real number, its polar form is \( 4(\cos 0 + i \sin 0) \).

Step 2: Find the Product \( z_1 z_2 \)

The product of two complex numbers in polar form is found by multiplying their moduli and adding their arguments. Thus, \( z_1 z_2 = (5\sqrt{2})(4)[\cos(\frac{\pi}{4} + 0) + i \sin(\frac{\pi}{4} + 0)] = 20\sqrt{2} [\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4})] \).

Step 3: Find the Quotient \( \frac{z_1}{z_2} \)

The quotient of two complex numbers in polar form is found by dividing their moduli and subtracting their arguments. Thus, \( \frac{z_1}{z_2} = \frac{5\sqrt{2}}{4}[\cos(\frac{\pi}{4} - 0) + i \sin(\frac{\pi}{4} - 0)] = \frac{5\sqrt{2}}{4}[\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}] \).

Step 4: Find the Quotient \( \frac{1}{z_1} \)

To find \( \frac{1}{z_1} \), we need to take the reciprocal of the polar form of \( z_1 \). The modulus becomes the reciprocal: \( \frac{1}{5\sqrt{2}} \) and the argument becomes its negative:\( \frac{1}{z_1} = \frac{1}{5\sqrt{2}}[\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4})] \).

Key Concepts

Polar Form

Complex numbers can be represented in polar form, which combines both their magnitude and direction. This form is particularly useful in multiplication and division. In polar form, a complex number is expressed as:

• The modulus, which represents the length of the vector in the complex plane (also known as magnitude).
• The argument, which is the angle the vector makes with the positive real axis.

For instance, if you have the complex number 5 + 5i, the polar form will make it easier to visualize and calculate its product and quotient with other complex numbers. This involves converting the rectangular form into polar coordinates by finding both the modulus and the argument, allowing for a clear representation of the original number.

Modulus

The modulus of a complex number, often denoted as \(r\), is its distance from the origin in the complex plane. It is calculated using the Pythagorean theorem. For a complex number in the form \(a + bi\), the modulus is \( \sqrt{a^2 + b^2} \).
For example, to find the modulus of \(z_1 = 5 + 5i\), use:\[r = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}\]This modulus serves as the "length" of the number from the origin and is an essential component when expressing numbers in polar form.
It is always a non-negative real number and provides valuable information about the number's magnitude.

Argument

The argument of a complex number is the angle measured in radians between the positive real axis and the line representing the complex number in the complex plane.
It helps in precisely locating where the number "points" from the origin. For the complex number \(a + bi\), the argument \(\theta\) is found using the formula \(\theta = \tan^{-1}(b/a)\).In the case of \(z_1 = 5 + 5i\), the argument is:\[\theta = \tan^{-1}(\frac{5}{5}) = \frac{\pi}{4}\]This shows that \(z_1\) forms an angle of \(45^\circ\) or \(\frac{\pi}{4}\) radians with the positive real axis.
Calculating the argument is crucial for working with complex numbers in polar form as it indicates direction.

Product of Complex Numbers

To find the product of two complex numbers expressed in polar form, you simply multiply their moduli and add their arguments.
This method leverages the property of polar forms that makes multiplication straightforward.For the given complex numbers \(z_1 = 5\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\) and \(z_2 = 4\cdot(\cos 0 + i \sin 0)\), their product is obtained as follows:

• Multiply their moduli: \((5\sqrt{2} \times 4) = 20\sqrt{2}\)
• Add their arguments: \(\frac{\pi}{4} + 0 = \frac{\pi}{4}\)

Thus, the product is:\[ 20\sqrt{2} (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\]This makes polar representation powerful for operations such as multiplication, offering a neat and concise multiplication rule that sums angles and scales lengths.