Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$4 \sqrt{3}+4 i$$

Step-by-Step Solution

Answer

The complex number in trigonometric form is \( 8\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right).\)

Step 1: Identify the Components

The given complex number is \( 4\sqrt{3} + 4i \). Identify the real part \( a = 4\sqrt{3} \) and the imaginary part \( b = 4 \).

Step 2: Calculate the Magnitude

The magnitude \( r \) of a complex number \( a + bi \) is given by \( r = \sqrt{a^2 + b^2} \). Here, \( r = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \).

Step 3: Find the Argument

The argument \( \theta \) is calculated using \( \tan(\theta) = \frac{b}{a} = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}} \). This is equivalent to \( \tan(\theta) = \tan(\frac{\pi}{6}) \). Hence, \( \theta = \frac{\pi}{6} \).

Step 4: Write in Trigonometric Form

The trigonometric form of the complex number is given by \( z = r(\cos(\theta) + i\sin(\theta)) \). Substituting the values, we get \( z = 8\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right) \).

Key Concepts

Real and Imaginary Components

Every complex number is made up of two basic parts: the real part and the imaginary part. For the complex number given, \( z = 4\sqrt{3} + 4i \), we identify the real and imaginary components as follows:

• The real component, \( a \), is the number that does not have the imaginary unit \( i \). In this case, it is \( 4\sqrt{3} \).
• The imaginary component, \( b \), is the coefficient of \( i \). Here, it equals \( 4 \).

These components can be visualized on the complex plane, where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers. Understanding these components is the first step in breaking down complex numbers and is vital for further steps like determining the magnitude and the argument.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as its modulus, measures the "size" or length of the vector representing the complex number on the complex plane. For a number \( z = a + bi \), the magnitude \( r \) is calculated using the formula:\[r = \sqrt{a^2 + b^2}\]In our example of \( 4\sqrt{3} + 4i \):

• First, compute \( (4\sqrt{3})^2 = 48 \).
• Then, \( 4^2 = 16 \).
• Add them together to get \( 48 + 16 = 64 \).
• Finally, take the square root to find \( r = \sqrt{64} = 8 \).

The magnitude reflects how far the point \( (a, b) \) is from the origin (0, 0) on the complex plane. It's a crucial component when converting a complex number into trigonometric form.

Argument of Complex Numbers

The argument of a complex number is the angle \( \theta \) that the line representing the complex number makes with the positive real axis. It places the complex number in its correct position within the polar coordinate system.To find \( \theta \), we use the tangent function:\[\tan(\theta) = \frac{b}{a}\]In this example:

• \( \tan(\theta) = \frac{4}{4\sqrt{3}} = \frac{1}{\sqrt{3}} \)

This is equivalent to \( \tan(\theta) = \tan\left(\frac{\pi}{6}\right) \), so \( \theta = \frac{\pi}{6} \).The argument is typically measured in radians and is important for positioning the complex number in its trigonometric form. Knowing \( \theta \) allows you to accurately describe the direction of the complex number as opposed to merely its magnitude.

Trigonometric Representation

Converting a complex number into its trigonometric form allows for expressing it in terms of its magnitude and direction. This representation is beneficial in various applications, such as multiplying and dividing complex numbers, because it simplifies these operations.The trigonometric form of a complex number \( z = a + bi \) is written as:\[z = r(\cos(\theta) + i\sin(\theta))\]Where:

• \( r \) is the magnitude.
• \( \theta \) is the argument.

For the complex number \( 4\sqrt{3} + 4i \), we found \( r = 8 \) and \( \theta = \frac{\pi}{6} \). Substituting these values gives:\[z = 8\left( \cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right) \right)\]The trigonometric form ties together the magnitude and direction into a single representation, allowing easier manipulation, especially for complex arithmetic operations.