Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -7 $$

Step-by-Step Solution

Answer

The complex number in trigonometric form is \(7(\cos \pi + i\sin \pi)\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(-7\), which can be written as: \(-7 + 0i\). This means the real part is \(-7\) and the imaginary part is \(0\).

Step 2: Calculate the Modulus

The modulus \(r\) of the complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). For this problem, the values are \(a = -7\) and \(b = 0\), so:\[r = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7\].

Step 3: Calculate the Argument \(\theta\)

The argument \(\theta\) is the angle from the positive x-axis to the line representing the complex number. For a purely real number \(-7\), lying on the negative x-axis, \(\theta\) is \(\pi\) radians. Thus, \(\theta = \pi\) because it lies on the negative real axis.

Step 4: Write in Trigonometric Form

The trigonometric form of a complex number is given by \(r(\cos \theta + i\sin \theta)\). Substituting our values, we get: \[7(\cos \pi + i\sin \pi)\].

Key Concepts

Trigonometric Form

The trigonometric form of complex numbers is a way to express a complex number using its modulus and argument. This form is particularly useful because it provides insight into the geometric properties of complex numbers, such as their angle and distance from the origin on the complex plane.
When converting a complex number into trigonometric form, we utilize the formula:

• Given a complex number in the form of \( a + bi \), the trigonometric form is \( r(\cos \theta + i\sin \theta) \)
• Where \( r \) is the modulus and \( \theta \) is the argument

For example, if we have the complex number \(-7\), which can be written as \(-7 + 0i\), its trigonometric form would be \( 7(\cos \pi + i\sin \pi) \) based on the step-by-step solution provided. This illustrates the number's position directly on the negative real axis, a distance of 7 units from the origin.

Modulus

The modulus of a complex number measures how far the number is from the origin in the complex plane.
It is denoted by \( r \) and calculated using the formula:

• \( r = \sqrt{a^2 + b^2} \)

where \( a \) is the real part and \( b \) is the imaginary part of the complex number \( a + bi \). The modulus is always a non-negative real number.
For the complex number \(-7 + 0i\), the modulus calculation is:

• \( r = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7 \)

This represents the distance of the number \(-7\) from the origin, confirming it lies 7 units away directly to the left (since it is negative).

Argument

The argument of a complex number, represented by \( \theta \), is the angle formed between the positive real axis and the line representing the complex number on the complex plane. It determines the direction of the number from the origin.
The argument can be calculated depending on the quadrant in which the complex number lies. For purely real numbers like \(-7\) which lie directly on the real axis, the determination of \( \theta \) is straightforward.
In our case, since \(-7\) is on the negative real axis, the argument is:

• \( \theta = \pi \)

This indicates a straight line in the negative direction of the x-axis, essentially pointing 180 degrees (or \( \pi \) radians) from the positive direction.

Real and Imaginary Parts

To express complex numbers in different forms, it's essential to first identify their real and imaginary components.
A complex number generally follows the form \( a + bi \), where:

• \( a \) is the real part
• \( b \) is the imaginary part

When dealing with the number \(-7\), it can be rewritten as \(-7 + 0i\). This makes it clear that:

• The real part is \(-7\)
• The imaginary part is \(0\)

Understanding these parts allows us to easily calculate both the modulus and the argument, simplifying the process of converting the number into its trigonometric form.