Problem 37 - Algebra and Trigonometry

Question

In Exercises \(37-44,\) find the product of the complex numbers. Leave answers in polar form. $$ \begin{array}{l} {z_{1}=6\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)} \\ {z_{2}=5\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)} \end{array} $$

Step-by-Step Solution

Answer

The product of the given complex numbers in polar form is \(30(\cos{70^{\circ}} + i\sin{70^{\circ}})\)

Step 1: Identify the magnitudes and angles

The magnitude of \(z_1\) is 6 and its angle is \(20^{\circ}\). Similarly, the magnitude of \(z_2\) is 5 and its angle is \(50^{\circ}\).

Step 2: Multiply the magnitudes and add the angles

Multiply the magnitudes of \(z_1\) and \(z_2\) to get the magnitude of the product. So, \(6 \times 5 = 30\). Add the angles of \(z_1\) and \(z_2\) to get the angle of the product. So, \(20^{\circ} + 50^{\circ} = 70^{\circ}\).

Step 3: Write the product in polar form

Now write the product of \(z_1\) and \(z_2\) in polar form, i.e., as a magnitude and an angle. The magnitude is 30 and the angle is \(70^{\circ}\). So, the product of \(z_1\) and \(z_2\) is \(30(\cos{70^{\circ}} + i\sin{70^{\circ}})\).

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number provides a different way to represent complex numbers, focusing on their magnitude and argument (the angle with the positive direction of the x-axis). A complex number, such as
\( z = a + bi \), can be expressed in polar form as
\( r(\cos \theta + i\sin \theta) \),
where \( r = \sqrt{a^2 + b^2} \) is the magnitude (or modulus) of the complex number and \( \theta \) is the argument (or angle), calculated as \( \text{atan2}(b, a) \). The functions \( \cos \) and \( \sin \) are the trigonometric cosine and sine with \( \theta \) expressed in radians or degrees.
In the given exercise, \( z_1 \) and \( z_2 \) are both given in polar form, where the magnitude and angle are explicitly stated, making it easy to visualize and multiply them as we will see in the next section.

Multiplying Complex Numbers

When multiplying complex numbers in polar form, we utilize their magnitudes and arguments:

• The magnitudes are multiplied together.
• The arguments (angles) are added.

This is rooted in the trigonometric addition formulas, which is why the polar form is so beneficial for multiplication of complex numbers. The resulting complex number has the new magnitude and angle derived from these operations.
For the given exercise, we apply these rules: multiply the magnitudes of \( z_1 \) and \( z_2 \) to obtain 30 (
\( 6 \times 5 \)) and add the angles to obtain \( 70^\circ \) (
\( 20^\circ + 50^\circ \)). Thus, the product is
\( 30(\cos{70^\circ} + i\sin{70^\circ}) \), a neat result that showcases the elegance of complex number multiplication in polar form.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool especially used for raising complex numbers to a power. The theorem states that for a complex number \( z = r(\cos \theta + i\sin \theta) \) and a positive integer \( n \), the nth power of the complex number can be represented as
\( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) \).
This is particularly useful for computing higher powers of complex numbers without having to multiply them out the long way. While not required for multiplying two complex numbers as in our exercise, understanding De Moivre's Theorem provides a foundation for more advanced operations with complex numbers and comes in handy when dealing with powers instead of just products.