Problem 54 - 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. $$ \begin{array}{l}{z_{1}=\sqrt{2}\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)} \\\ {z_{2}=3 \sqrt{2}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)}\end{array} $$

Step-by-Step Solution

Answer

The product is \(6\text{cis}135^\circ\) and the quotient is \(\frac{1}{3}\text{cis}15^\circ\).

Step 1: Identify Given Complex Numbers

We have two complex numbers given in trigonometric form:
\[ z_1 = \sqrt{2} (\cos 75^\circ + i \sin 75^\circ) \] \[ z_2 = 3\sqrt{2} (\cos 60^\circ + i \sin 60^\circ) \]

Step 2: Calculate the Product

Using the product of complex numbers in polar form: \( r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \)
\[ z_1 z_2 = \sqrt{2} \cdot 3\sqrt{2} \left( \cos(75^\circ + 60^\circ) + i \sin(75^\circ + 60^\circ) \right) = 6 (\cos 135^\circ + i \sin 135^\circ) \]

Step 3: Convert Product to Polar Form

The product \( z_1 z_2 \) is already in polar form with magnitude 6 and argument \( 135^\circ \):
\[ z_1 z_2 = 6\text{cis}135^\circ \]

Step 4: Calculate the Quotient

Using the quotient of complex numbers in polar form: \( \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)) \)
\[ \frac{z_1}{z_2} = \frac{\sqrt{2}}{3\sqrt{2}} \left( \cos(75^\circ - 60^\circ) + i \sin(75^\circ - 60^\circ) \right) = \frac{1}{3} \left( \cos 15^\circ + i \sin 15^\circ \right) \]

Step 5: Convert Quotient to Polar Form

The quotient \( \frac{z_1}{z_2} \) is in polar form with magnitude \( \frac{1}{3} \) and argument \( 15^\circ \):
\[ \frac{z_1}{z_2} = \frac{1}{3} \text{cis}15^\circ \]

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, especially useful in fields like engineering and physics. They are composed of two parts: a real part and an imaginary part. These numbers can be represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).
Complex numbers can represent more than just mathematical solutions; they also describe phenomena in electrical circuits or signal processing.

• Real part \(a\) handles the measurable physical quantity.
• Imaginary part \(bi\) often represents a transformation or phase shift.

Additionally, complex numbers can be expressed using different forms such as rectangular (Cartesian), polar, and exponential, each suitable for various applications.

Polar Coordinates

Polar coordinates offer a useful way to represent complex numbers, especially when dealing with multiplication and division. Unlike Cartesian coordinates that work with horizontal and vertical distances, polar coordinates utilize a radial distance and an angle from a fixed direction.
This method is particularly powerful in mathematics because it allows for a seamless interpretation of rotational and spiral patterns by using:

• An angle \(\theta\) (often in degrees or radians) that describes the direction of the point from the positive x-axis.
• A radius \(r\), which is the distance from the origin to the point.

In polar form, a complex number \(z\) is represented as \(r (\cos \theta + i \sin \theta)\). This alternative form proves especially efficient for multiplication and division, as shown in our exercise with \(z_1\) and \(z_2\).

Trigonometric Form

The trigonometric form of complex numbers is closely associated with polar coordinates. It expresses a complex number in terms of its magnitude and angle.
When we express a complex number like \(z_1\) or \(z_2\) using trigonometric form, it becomes easier to carry out operations such as multiplication and division because this form directly leverages angle identities.

• The magnitude is the distance from the origin to the point, expressed as \(r\).
• The angle \(\theta\) is the counterclockwise rotation from the positive x-axis.

The format \(r(\cos \theta + i \sin \theta)\) simplifies arithmetic calculations by converting them into operations on magnitudes and angles. This form is particularly efficient for finding products like \(z_1 z_2\) or quotients like \(\frac{z_1}{z_2}\), as it simplifies to calculating cosine and sine of summed or subtracted angles.

Angle Sum Identity

The angle sum identity is a trigonometric property that simplifies calculations involving the sum of angles. It is particularly useful in converting products and quotients of complex numbers from the trigonometric form, as it helps combine two angles into a single term.
The well-known identities are:

• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

These identities allow us to perform operations on complex numbers easily. For example, to find the product \(z_1 z_2\), we apply these identities to combine the angles \(75^\circ\) and \(60^\circ\) into a single angle \(135^\circ\). It also supports the process of expressing results in polar form, maintaining both magnitude and resulting angles.