Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$-3-3 i \sqrt{3}$$

Step-by-Step Solution

Answer

The trigonometric form is \( 6(\cos 240^{\circ} + i\sin 240^{\circ}) \).

Step 1 - Identify the Real and Imaginary Parts

First, identify the real and imaginary parts of the complex number. In this case, the complex number is \(-3 - 3i\sqrt{3} \), where the real part is \(a = -3\) and the imaginary part is \(b = -3\sqrt{3}\).

Step 2 - Calculate the Magnitude

To find the magnitude \(r\) of the complex number, use the formula \( r = \sqrt{a^2 + b^2} \). Substitute \(a\) and \(b\): \\[r = \sqrt{(-3)^2 + (-3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6. \\]

Step 3 - Determine the Argument \(\theta\)

To find \(\theta\), use the formula \(\tan \theta = \frac{b}{a}\). Substituting values: \\[\tan \theta = \frac{-3\sqrt{3}}{-3} = \sqrt{3}. \\]The angle \(\theta\) for \(\tan\theta = \sqrt{3}\) is \(60^{\circ}\), but since the complex number is in the third quadrant, adjust the angle to \( \theta = 240^{\circ}\).

Step 4 - Write the Complex Number in Trigonometric Form

With \(r = 6\) and \(\theta = 240^{\circ}\), express the complex number in trigonometric form: \\[6\left(\cos 240^{\circ}+i\sin 240^{\circ}\right).\\]

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are typically expressed in the form of \( a + bi \), where \( a \) is the real number component and \( b \) is the imaginary number component, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).
For example, the number \(-3 - 3i\sqrt{3}\) is a complex number where \( a = -3 \) and \( b = -3\sqrt{3} \).
Complex numbers can be visualized on a complex plane, similar to the Cartesian plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. They are useful in many fields like engineering and physics for solving equations that can't be solved using only real numbers.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as the modulus, is a measure of the size, or length, of the vector representing the complex number on the complex plane. It is denoted as \( r \) and can be 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.

• For instance, for the complex number \(-3 - 3i\sqrt{3}\), the magnitude is calculated as:
  \[ r = \sqrt{(-3)^2 + (-3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6 \]

This represents the distance of the point \(-3 - 3i\sqrt{3}\) from the origin (0,0) on the complex plane. The magnitude is always a non-negative real number.

Argument of Complex Numbers

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. It's usually represented by \( \theta \) and can be found using the tangent function:
\[ \tan \theta = \frac{b}{a} \]
Since the tangent function can have the same values for angles in different quadrants, it's crucial to identify the correct quadrant for the complex number.
For the complex number \(-3 - 3i\sqrt{3}\), we calculate:
\[ \tan\theta = \frac{-3\sqrt{3}}{-3} = \sqrt{3} \]
This would commonly correspond to an angle of \(60^{\circ}\). However, because the number is in the third quadrant, the correct angle is \( \theta = 240^{\circ} \).
The argument helps to convert complex numbers into trigonometric form which is very useful in complex calculations.

Third Quadrant in Trigonometry

The trigonometric plane is divided into four quadrants, each representing a different combination of signs for the real and imaginary components when dealing with complex numbers.
The third quadrant corresponds to the region where both the real and imaginary parts of the complex numbers are negative.
In this quadrant, angles range from \(180^{\circ}\) to \(270^{\circ}\).
When a complex number like \(-3 - 3i\sqrt{3}\) lies in the third quadrant:

• The real part \( a = -3 \) is negative.
• The imaginary part \( b = -3\sqrt{3} \) is also negative.

To determine the angle \( \theta \) in this quadrant, we adjust our initial angle calculation by adding \( 180^{\circ} \) to account for the third quadrant position. Thus an angle of \(60^{\circ}\) becomes \(240^{\circ}\). Understanding the quadrant helps ensure accurate trigonometric transformations.