Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ \sqrt{3}-i $$

Step-by-Step Solution

Answer

The trigonometric form of \( \sqrt{3} - i \) is \( 2 \left( \cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6} \right) \).

Step 1: Identify Real and Imaginary Parts

The complex number given is \( \sqrt{3} - i \). Here, \( \sqrt{3} \) is the real part, and \( -1 \) is the imaginary part.

Step 2: Calculate the Magnitude

To find the magnitude \( r \) of the complex number \( \sqrt{3} - i \), use the formula: \[ r = \sqrt{a^2 + b^2} \]Substitute \( a = \sqrt{3} \) and \( b = -1 \):\[ r = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

Step 3: Determine the Argument

The argument \( \theta \) of the complex number is found using the formula:\[ \theta = \tan^{-1}\left( \frac{b}{a} \right) \]Substitute \( a = \sqrt{3} \) and \( b = -1 \):\[ \theta = \tan^{-1}\left( \frac{-1}{\sqrt{3}} \right) = \tan^{-1}\left( -\frac{1}{\sqrt{3}} \right) \]This corresponds to an angle of \( -\frac{\pi}{6} \), which is equivalent to \( \frac{11\pi}{6} \) in the range \( [0, 2\pi) \).

Step 4: Write the Trigonometric Form

Express the complex number \( \sqrt{3} - i \) in trigonometric form using:\[ z = r (\cos \theta + i \sin \theta) \]Substitute \( r = 2 \) and \( \theta = \frac{11\pi}{6} \):\[ \sqrt{3} - i = 2 \left( \cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6} \right) \]

Key Concepts

Magnitude of Complex Numbers

In the realm of complex numbers, the magnitude (or modulus) represents the distance of the complex number from the origin in the complex plane. It is similar to finding the length of the hypotenuse in a right-angled triangle made from the real and imaginary components.
To find the magnitude of a complex number in the form \( a + bi \), you use the formula:

• \( r = \sqrt{a^2 + b^2} \)

For the given complex number \( \sqrt{3} - i \), the real part is \( \sqrt{3} \), and the imaginary part is \( -1 \).
Substitute these values into the formula:

• \( r = \sqrt{(\sqrt{3})^2 + (-1)^2} \)
• \( r = \sqrt{3 + 1} = \sqrt{4} = 2 \)

Here, \( r = 2 \) is the magnitude, indicating our complex number is 2 units away from the origin.

Argument of Complex Numbers

The argument of a complex number is the angle the number makes with the positive direction of the real axis. Finding this angle helps in expressing the complex number in polar or trigonometric form.
The argument \( \theta \) can be found with the formula:

• \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \)

where \( a \) and \( b \) are the real and imaginary parts, respectively.
For \( \sqrt{3} - i \), substitute \( a = \sqrt{3} \) and \( b = -1 \):

• \( \theta = \tan^{-1}\left( -\frac{1}{\sqrt{3}} \right) \)

This calculation gives \( -\frac{\pi}{6} \), which needs adjusting to fit into the standard range of \([0, 2\pi)\).
Adding \( 2\pi \) for the correct angle, we get \( \frac{11\pi}{6} \).

Polar Coordinates

Polar coordinates provide a way of expressing complex numbers through magnitude and angle instead of their real and imaginary parts.
This is particularly useful because it reflects both the position and orientation of a complex number in the complex plane in a straightforward way.

• The magnitude tells us the distance from the origin.
• The angle (or argument) describes the direction from the origin.

For \( \sqrt{3} - i \), converted into polar coordinates:

• The magnitude, \( r = 2 \).
• The argument, \( \theta = \frac{11\pi}{6} \).

In trigonometric form, this is represented as:

• \( z = r(\cos \theta + i \sin \theta) \)
• \( z = 2\left( \cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6} \right) \)

This expression encapsulates both the magnitude and the direction of the complex number.

Real and Imaginary Parts

Every complex number can be broken down into two key parts: the real part and the imaginary part. This breakdown is essential for performing arithmetic operations and simplifying expressions.
The real and imaginary components are expressed as \( a \) and \( b \) in a complex number \( a + bi \):

• The real part is \( a \), which is the coefficient of the real number.
• The imaginary part is \( b \), the coefficient of \( i \), the imaginary unit.

For the complex number \( \sqrt{3} - i \):

• The real part is \( \sqrt{3} \).
• The imaginary part is \( -1 \).

Understanding these parts allows us to move between different forms of complex numbers, such as cartesian and polar, and enables us to use operations like addition, subtraction, and finding magnitudes.