Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 15 $$

Step-by-Step Solution

Answer

The trigonometric form is \( 15(\cos 0 + i \sin 0) \).

Step 1: Identify the complex number

The complex number given is 15. Since there is no imaginary part, this is a real number.

Step 2: Write the complex number in rectangular form

A complex number is usually written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Here, it is \( 15 + 0i \).

Step 3: Find the modulus of the complex number

The modulus \( r \) of a complex number \( a + bi \) is given by \( r = \sqrt{a^2 + b^2} \). For 15,\[ r = \sqrt{15^2 + 0^2} = \sqrt{225} = 15. \]

Step 4: Determine the argument \( \theta \)

For a real number on the positive side of the real axis, the angle \( \theta \) from the positive real axis is 0. Hence, \( \theta = 0 \).

Step 5: Express in trigonometric form

The trigonometric form of a complex number is \( r(\cos \theta + i \sin \theta) \). Substituting the values obtained, \[ 15(\cos 0 + i \sin 0) = 15 \times 1 = 15. \]

Key Concepts

Complex Number Representation

Complex numbers are an extension of the real numbers and include an imaginary part. A complex number is usually represented as \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part, with \( i \) being the imaginary unit defined as \( \sqrt{-1} \). This representation allows complex numbers to express quantities that cannot be described using only real numbers.

• The real part \( a \) is a real number.
• The imaginary part \( b \) is also a real number, but it is multiplied by \( i \) making it imaginary.

A complex number can be illustrated on the complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part.

Modulus of Complex Numbers

The modulus of a complex number is a measure of its magnitude or size and is denoted by \( |z| \) for a complex number \( z = a + bi \).
The modulus is calculated using the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

For example, the modulus of 15 can be calculated as follows:
Given \( 15 + 0i \),

• \( |15| = \sqrt{15^2 + 0^2} = \sqrt{225} = 15 \)

The modulus provides the distance of the complex number from the origin (0,0) on the complex plane. A real number will have its modulus as the absolute value of the number itself since the imaginary part is zero.

Argument of Complex Numbers

The argument of a complex number refers to the angle made by the line representing the complex number on the complex plane with the positive real axis. It is often denoted by \( \theta \) and measured in radians.
For a complex number \( a + bi \), the argument can be determined using trigonometry. However, for real numbers (where the imaginary part is zero), the task is simplified:

• Positive real numbers have an argument \( \theta = 0 \).
• Negative real numbers have an argument \( \theta = \pi \).

In our example of the complex number 15:

• Since 15 is a positive real number, \( \theta = 0 \).

Rectangular Form of Complex Numbers

The rectangular form of a complex number expresses it as \( a + bi \), where:

• The term \( a \) represents the horizontal (real) component.
• The term \( bi \) represents the vertical (imaginary) component.

In rectangular form, a complex number is visualized on the complex plane as a point or vector from the origin to the position \( (a, b) \).
For our example, with 15 as the complex number, the rectangular form is \( 15 + 0i \). This means:

• The vector lies entirely on the real axis.
• No contribution from the imaginary axis as \( b = 0 \).

Understanding this form is crucial for converting complex numbers into other forms, such as polar (trigonometric) form.