Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$-5-5 i$$

Step-by-Step Solution

Answer

The trigonometric form is \(5\sqrt{2}(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4})\).

Step 1: Identify the Complex Number

The given complex number is \(-5 - 5i\).Here, the real part is \(-5\), and the imaginary part is also \(-5\).

Step 2: Calculate the Modulus

The modulus \(r\) of a complex number \(a + bi\) is calculated using the formula \[ r = \sqrt{a^2 + b^2} \].For \(-5-5i\), this becomes \[ r = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \].

Step 3: Determine the Argument

The argument \(\theta\) is calculated using \[ \tan\theta = \frac{b}{a} \].Substitute \(a = -5\) and \(b = -5\):\[ \tan\theta = \frac{-5}{-5} = 1 \].Thus, \(\theta = \frac{3\pi}{4} + \pi = \frac{7\pi}{4}\) since the complex number is in the third quadrant.

Step 4: Write in Trigonometric Form

The trigonometric form of a complex number is written as \[ r(\cos\theta + i\sin\theta) \].Substitute \(r = 5\sqrt{2}\) and \(\theta = \frac{7\pi}{4}\):The complex number in trigonometric form is \[ 5\sqrt{2}(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}) \].

Key Concepts

Trigonometric Form

The trigonometric form of a complex number is a way of expressing complex numbers using their geometric properties. Instead of just showing the real and imaginary parts, as is common in the standard form \(a + bi\), the trigonometric form emphasizes the magnitude (modulus) and direction (argument) of the number in the complex plane.
This form is particularly useful for multiplication and division, as well as for finding powers and roots of complex numbers.

In the trigonometric form, a complex number is expressed as:

• \( r(\cos \theta + i\sin \theta) \)

Where:

• \( r \) is the modulus of the complex number.
• \( \theta \) is the argument (or angle) of the complex number.

For the given complex number \(-5 - 5i\), the trigonometric form is \( 5\sqrt{2}(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}) \). By converting complex numbers into this form, complex mathematical operations often become simpler.

Modulus of Complex Numbers

The modulus of a complex number is a measure of its size or magnitude, often denoted by \( r \). It represents the distance of the complex number from the origin on the complex plane.
The modulus can be thought of similar to calculating the hypotenuse of a right triangle where one side is the real part and the other side is the imaginary part.

The formula to calculate the modulus for a complex number \( a + bi \) is:

• \( r = \sqrt{a^2 + b^2} \)

For the complex number \(-5 - 5i\), this becomes:

• \( r = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \)

This modulus shows that the given complex number is \( 5\sqrt{2} \) units away from the origin, providing the necessary magnitude component for its trigonometric representation.

Argument of Complex Numbers

The argument of a complex number, often symbolized as \( \theta \), is the angle that the line representing the complex number makes with the positive real axis in the complex plane.
This angle is an important aspect of the trigonometric form as it dictates the direction of the complex number from the origin.
The calculation of the argument depends on the quadrant in which the complex number lies.

The basic formula for calculating the argument is:

• \( \tan\theta = \frac{b}{a} \)

For \(-5 - 5i\), substituting \( a = -5 \) and \( b = -5 \) gives:

• \( \tan\theta = 1 \)

Since the complex number lies in the third quadrant, the angle \( \theta \) can be determined as:

• \( \theta = \frac{3\pi}{4} + \pi = \frac{7\pi}{4} \)

This argument provides the directional component for the trigonometric form, ensuring the complex number is accurately represented on the complex plane.