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Chapter 7: Problem 31

Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\) $$12$$

Short Answer

Expert verified
The trigonometric form is \(12(\cos 0 + i\sin 0)\).

Step by step solution

01

Identify the modulus

The modulus of a complex number \(a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). Here, the complex number is \(12 + 0i\). So, \(|z| = \sqrt{12^2 + 0^2} = 12\).
02

Determine the argument

The argument \(\theta\) of a complex number in the form \(a + 0i\) where \(a > 0\) is 0, since the angle from the positive x-axis to the point on the real axis is 0 radians.
03

Express in trigonometric form

The trigonometric form of a complex number is given by \(z = r(\cos \theta + i\sin \theta)\). Here, the modulus \(r = 12\) and the argument \(\theta = 0\). Therefore, the trigonometric form is: \(12(\cos 0 + i\sin 0)\) or simply \(12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Form
The trigonometric form of a complex number expresses the number in terms of its modulus and argument. When we represent complex numbers like this, it allows us to understand the graphical interpretation better. In trigonometric form, a complex number \( z \) is given by:
  • \( z = r(\cos \theta + i\sin \theta) \)
  • Here, \( r \) is the modulus, and \( \theta \) is the argument.
For the exercise, the complex number \( 12 + 0i \) is on the real axis. Its modulus is 12, and its argument, the angle with respect to the positive x-axis, is 0 radians.
The trigonometric form visually shows how a complex number can be split into real and imaginary components using circular functions \( \cos \theta \) and \( \sin \theta \).
Modulus of Complex Numbers
The modulus of a complex number is a measure of its size or length in the complex plane. For a complex number \( a + bi \), the modulus is calculated as:
  • \( |z| = \sqrt{a^2 + b^2} \)
In the given exercise, the complex number \( 12 + 0i \) is purely real. Thus, its modulus is simply the absolute value of the real part, which is 12.
This modulus can be thought of as the distance from the origin to the point \((12, 0)\) on the complex plane. This value is crucial because it tells us how far the complex number is from the origin.
Argument of Complex Numbers
The argument of a complex number is the angle \( \theta \) formed with the positive x-axis in the complex plane. It describes the direction of the number from the origin. For any complex number \( a + bi \), the argument is given by:
  • \( \theta = \text{atan2}(b, a) \) when \( a eq 0 \)
  • If \( a > 0 \) and \( b = 0 \), \( \theta = 0 \) radians
In this particular problem, the complex number is \( 12 + 0i \). Since \( a = 12 \) and \( b = 0 \), and \( a > 0 \), the argument is 0 radians.
The argument is essential because it gives us the direction of the complex number's vector in the plane, helping to visualize and understand its placement relative to the origin.

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