Problem 46 - 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 \(0^{\circ} \leq \theta<360^{\circ}\) $$-2+2 i \sqrt{3}$$

Step-by-Step Solution

Answer

The complex number is \(4(\cos 120^\circ + i \sin 120^\circ)\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(-2 + 2i\sqrt{3}\). Here, the real part is \(-2\) and the imaginary part is \(2\sqrt{3}\). We will use these components in the next steps.

Step 2: Calculate the Magnitude

The magnitude or modulus \(r\) of the complex number is calculated using the formula: \[r = \sqrt{a^2 + b^2}\]where \(a = -2\) and \(b = 2\sqrt{3}\). Substitute these values into the formula:\[r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4\]So, the magnitude \(r\) is \(4\).

Step 3: Determine the Argument

The argument \(\theta\) is found using the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]where \(b = 2\sqrt{3}\) and \(a = -2\). First, calculate the tangent:\[\tan\theta = \frac{2\sqrt{3}}{-2} = -\sqrt{3}\]Since the complex number is in the second quadrant (negative real and positive imaginary), the reference angle for \(-\sqrt{3}\) is\(60^\circ\), so \(\theta = 180^\circ - 60^\circ = 120^\circ\).

Step 4: Write in Trigonometric Form

Now that we have \(r = 4\) and \(\theta = 120^\circ\), we can write the complex number in its trigonometric form:\[4(\cos 120^\circ + i \sin 120^\circ)\]This is the desired trigonometric form of the given complex number.

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical elements that combine real and imaginary components. A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part multiplied by the imaginary unit \(i\). The imaginary unit \(i\) is defined as the square root of -1, giving rise to unique arithmetic properties. For example, in the complex number \(-2 + 2i\sqrt{3}\), \(-2\) is the real part and \(2\sqrt{3}\) is the imaginary part. Understanding this structure allows us to work with complex numbers in various mathematical operations.

Complex numbers are used extensively in fields such as engineering, physics, and applied mathematics because they can represent phenomena such as electromagnetic waves and quantum mechanics. They provide a more comprehensive way to solve equations that do not have real solutions.

Magnitude of Complex Numbers

The magnitude of a complex number provides us with a measure of its size or distance from the origin in the complex plane. To find the magnitude, we use the formula:

• \(r = \sqrt{a^2 + b^2}\)

where \(a\) and \(b\) are the real and imaginary parts respectively. For example, with \(-2 + 2i\sqrt{3}\), we calculate the magnitude as follows:

• \(r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4\)

This calculation shows that the magnitude of our complex number is 4.

This magnitude is useful in determining how far the number is from the origin when plotted on the complex plane, functioning similarly to the modulus of a vector.

Argument of Complex Numbers

The argument of a complex number is the angle it forms with the positive real axis on the complex plane. We use the formula

• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\)

where \(b\) is the imaginary part, and \(a\) is the real part. For our example \(-2 + 2i\sqrt{3}\), we calculate:

• \(\tan \theta = \frac{2\sqrt{3}}{-2} = -\sqrt{3}\)

Since the number is in the second quadrant, where the real part is negative, and the imaginary part is positive, we adjust the angle. The reference angle for \(-\sqrt{3}\) is \(60^\circ\). Therefore:

• \(\theta = 180^\circ - 60^\circ = 120^\circ\)

The argument helps us understand the direction of the complex number vector in the complex plane.

Second Quadrant

The complex plane is divided into four quadrants, similar to the Cartesian coordinate system. The second quadrant is characterized by negative real numbers and positive imaginary numbers. In this context, for the complex number \(-2 + 2i\sqrt{3}\), it falls in the second quadrant because its real part \(-2\) is negative, and its imaginary part \(2\sqrt{3}\) is positive.

The angle for any complex number in the second quadrant can be determined by subtracting the reference angle from \(180^\circ\). This characteristic is crucial when determining the argument of a complex number that lies in this quadrant.

Understanding the positioning within the quadrants helps comprehend how complex numbers are distributed and assist in translating them to trigonometric or exponential forms easily.