Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=7, \quad z_{2}=3+5 i $$

Step-by-Step Solution

Answer

\( z_1 z_2 = 7\sqrt{34}(\cos 59.04^\circ + i \sin 59.04^\circ) \); \( \frac{z_1}{z_2} = \frac{7}{\sqrt{34}}(\cos(-59.04^\circ) + i \sin(-59.04^\circ)) \).

Step 1: Write Complex Numbers in Trigonometric Form

Firstly, let's express the complex numbers in trigonometric form.For \(z_1 = 7\), it's a real number, and can be written as \(z_1 = 7(\cos 0 + i \sin 0)\).For \(z_2 = 3 + 5i\), calculate the modulus \(|z_2|\) as follows: \[ |z_2| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]Next, find the argument \(\theta\) using \(\tan \theta = \frac{5}{3}\). \(\theta\) is approximately \(59.04^\circ\).Thus, the trigonometric form is \(z_2 = \sqrt{34}(\cos 59.04^\circ + i \sin 59.04^\circ)\).

Step 2: Multiply the Complex Numbers

To find the product \(z_1 z_2\), multiply the moduli and add the arguments.The modulus of the product:\[ |z_1| \cdot |z_2| = 7 \cdot \sqrt{34} = 7\sqrt{34} \]The argument of the product:\[ 0^\circ + 59.04^\circ = 59.04^\circ \]So, the product \(z_1 z_2\) in trigonometric form is \[ 7\sqrt{34}(\cos 59.04^\circ + i \sin 59.04^\circ) \].

Step 3: Divide the Complex Numbers

To find the quotient \(\frac{z_1}{z_2}\), divide the moduli and subtract the arguments.The modulus of the quotient:\[ \frac{|z_1|}{|z_2|} = \frac{7}{\sqrt{34}} \]The argument of the quotient:\[ 0^\circ - 59.04^\circ = -59.04^\circ \]Thus, the quotient \(\frac{z_1}{z_2}\) in trigonometric form is \[ \frac{7}{\sqrt{34}}(\cos(-59.04^\circ) + i \sin(-59.04^\circ)) \].

Key Concepts

Understanding Complex Number Multiplication

When dealing with complex numbers, multiplication can seem tricky at first, but it becomes clearer with the trigonometric form. Here’s how you can approach it:

To multiply two complex numbers in trigonometric form, you need to multiply their moduli (the plural of modulus) and add their arguments.

Here’s a simple breakdown of the process:

• **Multiply Moduli**: Take the modulus of each complex number and multiply them.
• **Add Arguments**: Find the argument of each complex number and add them.
• **Form a New Complex Number**: With the new modulus and argument, form the trigonometric expression.

By using this approach, multiplying complex numbers becomes systematic and easy. Remember, the formula in trigonometric form for products is:\[ |z_1z_2| = |z_1||z_2| \]\[ \text{arg}(z_1z_2) = \text{arg}(z_1) + \text{arg}(z_2) \]This makes it easy to multiply and find new positions for complex numbers on the complex plane.

Simplifying Complex Number Division

Dividing complex numbers can be simplified by using the trigonometric form. This form breaks down the process into a straightforward series of steps.

Here’s how you can do it:

• **Divide the Moduli**: Divide the modulus of the numerator by the modulus of the denominator.
• **Subtract Arguments**: Subtract the argument of the denominator from the argument of the numerator.
• **Put Together the New Form**: Combine the new modulus and argument to put the result in trigonometric form.

By doing this, the division becomes clear and manageable, avoiding the complications that might come with traditional division methods.

The trigonometric formulas for division are:
\[ \frac{|z_1|}{|z_2|} \]
\[ \text{arg}\left(\frac{z_1}{z_2}\right) = \text{arg}(z_1) - \text{arg}(z_2) \]
This lets you easily divide complex numbers and understand their magnitude and direction on the complex plane.

Exploring Modulus and Argument of Complex Numbers

The modulus and argument are fundamental in understanding the trigonometric form of complex numbers. Here’s what they represent and how to calculate them:

• **Modulus**: This is the distance from the origin to the point representing the complex number on the complex plane. It’s found using the formula \(|z| = \sqrt{a^2 + b^2}\), where \(a\) is the real part, and \(b\) is the imaginary part.
• **Argument**: This is the angle the line (from the origin to the point) makes with the positive real axis. It provides the direction of the complex number and can be found using the tangent function: \( \tan \theta = \frac{b}{a} \).

These aspects are crucial because together they allow you to express a complex number in its trigonometric form, giving valuable insights about its position and orientation on the complex plane. The trigonometric form is:\[ z = |z| (\cos \theta + i \sin \theta) \]