Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$\sqrt{2} \text { cis } \pi$$

Step-by-Step Solution

Answer

The rectangular form is \( -\sqrt{2} + 0i \).

Step 1: Understanding the 'cis' notation

The 'cis' notation represents a complex number in polar form as \( r \text{ cis } \theta = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

Step 2: Identify the components

For the given complex number \( \sqrt{2} \text{ cis } \pi \), the modulus \( r = \sqrt{2} \) and the argument \( \theta = \pi \).

Step 3: Calculate the cosine and sine

Next, find \( \cos \pi \) and \( \sin \pi \). We know from trigonometric identities that \( \cos \pi = -1 \) and \( \sin \pi = 0 \).

Step 4: Substitute into the rectangular form

Substitute the values back into the rectangular form equation: \( \sqrt{2}(\cos \pi + i \sin \pi) = \sqrt{2}(-1) + \sqrt{2}(0)i = -\sqrt{2} + 0i \).

Step 5: Simplify the expression

The simplified rectangular form of the complex number is \( -\sqrt{2} \). The imaginary part is zero, so the expression remains \( -\sqrt{2} + 0i \).

Key Concepts

Rectangular Form

Complex numbers can be expressed in different forms, and one of the most straightforward is the rectangular form. In this representation, a complex number is written as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part multiplied by the imaginary unit \(i\). This form is useful as it easily allows you to understand the real and imaginary components directly.
For example, if you have a complex number such as \(-\sqrt{2} + 0i\), like in the original exercise, you can quickly identify the real part, \(-\sqrt{2}\), and the imaginary part, which is \(0\) in this case. This is handy because when the imaginary part is zero, the number is purely real.

• Easy to identify real and imaginary parts: \(a = -\sqrt{2}, b = 0\).
• Useful for arithmetic operations such as addition or subtraction.

Understanding rectangular form is essential because it serves as the foundation for many operations in complex number calculations, making it a crucial concept for students to grasp.

Polar Form

The polar form of a complex number provides an alternative way to express the number using its magnitude and angle. In this form, a complex number is written as \(r(\cos \theta + i \sin \theta)\), sometimes also shown as \(r \text{ cis } \theta\). Here, \(r\) is the modulus or the distance from the origin, and \(\theta\) is the argument or angle from the positive x-axis.
This form is particularly useful in cases involving multiplication, division, and finding powers or roots of complex numbers because of the simplicity of handling exponents and trigonometric properties.

• \(r\): Modulus, representing the length of the vector.
• \(\theta\): Argument, the angle of rotation from the positive x-axis.

Polar form allows for a comprehensive understanding of complex numbers, especially in geometry and physics, where visualizing magnitude and direction can be critical. Our example, \(\sqrt{2} \text{ cis } \pi\), translates into the rectangular form as \(-\sqrt{2} + 0i\) by converting the trigonometric components of the angle \(\pi\).

Trigonometric Identities

Understanding trigonometric identities is crucial when working with complex numbers in polar form. That's because converting between polar and rectangular forms often involves calculating sine and cosine values. Important trigonometric identities include:

• \(\cos(\pi) = -1\)
• \(\sin(\pi) = 0\)

These identities help in calculating the exact rectangular form of complex numbers. In our original exercise, we used these identities to convert \(\sqrt{2} \text{ cis } \pi\) from polar to rectangular form. Thanks to \(\cos(\pi)\) and \(\sin(\pi)\), we could determine that the number is \(-\sqrt{2} + 0i\).
In mathematics, trigonometric identities are tools used not only for simplification but also for solving equations involving angles. These identities are vital for anyone dealing with wave functions, periodic phenomena, and rotation problems, making them essential knowledge for students in both pure and applied mathematics.