Problem 62 - Precalculus with Limits

Question

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(3i(1\ -\ \sqrt{2}i)\)

Step-by-Step Solution

Answer

The result of the operation in both forms is \(-18 - 18i\), which means our trigonometric form correctly matched our standard form.

Step 1: Trigonometric Form

To express \(3i(1\ -\ \sqrt{2}i)\) in trigonometric form, find its magnitude \(r\) and angle \(\Theta\). Start by express the complex number in standard form \(a+bi\), which is \(3i+3\sqrt{2}i^2\). Remembering that \(i^2 = -1\), we can simplify to \(-3\sqrt{2}+3i\). Then find the magnitude and angle using the formulas \(r=\sqrt{(-3\sqrt{2})^2+3^2} = 3\sqrt{3}\) and \(\Theta = \arctan(-3/\sqrt{2}) = -120^\circ\). The equivalent trigonometric form is \(r(\cos\Theta + i\sin\Theta)\), which gives \(3\sqrt{3}(\cos(-120^\circ) + i\sin(-120^\circ))\)

Step 2: Perform Operation Using Trigonometric Form

As the operation is multiplication in form, we can simplify the operation using the rule \(r_1r_2(\cos(\Theta_1 + \Theta_2) + i\sin(\Theta_1 + \Theta_2))\). The complex number \(3\sqrt{3}(\cos(-120^\circ) + i\sin(-120^\circ))\) multiplied by itself gives \(9(\cos(-240^\circ) + i\sin(-240^\circ))\).

Step 3: Perform Operation Using Standard Form

Performing the operation \(3i(1 - \sqrt{2}i) * 3i(1 - \sqrt{2}i)\) in standard form gives \(-18 - 18i\). The real part of the operation in standard form should match the \(\cos\) component of the result in trigonometric form and the imaginary part should match the \(\sin\) component.

Key Concepts

Complex Numbers

In mathematics, complex numbers are an extension of real numbers and can be understood as points in a plane. A complex number is typically written in standard form as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). A real-component \( a \) represents the horizontal axis (also known as the real axis), and the imaginary component \( b \) represents the vertical axis (known as the imaginary axis) in the complex plane.

The beauty of complex numbers lies in their versatility in various mathematical contexts, including solving equations that have no real solutions. By introducing an imaginary dimension, it becomes possible to perform a full range of arithmetic operations and facilitate deeper understanding of algebraic structures.

Representations of Complex Numbers

Aside from the standard form, complex numbers can also be expressed in trigonometric form, which is particularly useful when dealing with multiplication or division. In trigonometric form, a complex number is written as \( r(cos\theta + i\text{sin}\theta) \), where \( r \) is the magnitude of the complex number, and \( \theta \) is the angle it forms with the positive real axis, measured counterclockwise. This representation showcases the geometric interpretation of complex numbers and enables simple calculations with their angles and magnitudes.

De Moivre's Theorem

De Moivre's theorem is a powerful tool in complex number arithmetic, particularly when raising complex numbers to powers or extracting roots. It states that for any real number \( \theta \) and any positive integer \( n \), the following relationship holds: \( (cos\theta + i\text{sin}\theta)^n = cos(n\theta) + i\text{sin}(n\theta) \).

This theorem allows us to easily compute the powers of a complex number represented in trigonometric form without the need for multiplying it out in standard form. It is especially useful when dealing with higher powers, as it simplifies the process significantly.

Using De Moivre's Theorem

To apply De Moivre's Theorem, simply raise both the cosine and sine parts to the power \( n \) and multiply the angle \( \theta \) by \( n \). This method is not only simpler but also helps avoid errors that may occur when dealing with complex numbers in their standard form. It's a technique that underscores the interconnectedness of algebra and geometry in complex analysis.

Operations With Complex Numbers

Performing operations with complex numbers can be done in both their standard and trigonometric forms. When using standard form \( a + bi \), we perform addition, subtraction, and multiplication regularly, following algebraic rules and treating \( i \) as a variable.

Multiplication and Division in Trigonometric Form

However, multiplication and division are often more intuitive in trigonometric form, where we can simply multiply the magnitudes and add or subtract the angles, respectively. This process is reminiscent of polar coordinates and provides a straightforward computational method, particularly for repetitive multiplication or division.

Switching Between Forms

One key aspect of mastering operations with complex numbers is understanding when and how to convert between standard form and trigonometric form. The solution given in the original exercise showcases how operations are simplified using the trigonometric form and verifies the result in standard form, illustrating the conceptual unity between the two forms. Remember that the real part of the result should correspond to the cosine term, and the imaginary part should correspond to the sine term in trigonometric form.