Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{-i}{1+i}$$

Step-by-Step Solution

Answer

\(-\frac{1}{2} - i\frac{1}{2}\)

Step 1: Convert Numerator to Trigonometric Form

The numerator is \(-i\). Represent it as a point in the complex plane: it has real part 0 and imaginary part -1. This corresponds to the point (0, -1) in the Cartesian plane.The modulus (or absolute value) is \(|-i| = \sqrt{0^2 + (-1)^2} = 1\).The angle \( \theta \) for \(-i\) corresponds to 270° or \( \frac{3\pi}{2} \) radians (as it lies along the negative imaginary axis).Thus, the trigonometric form of \(-i\) is \(1 \times (\cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right))\).

Step 2: Convert Denominator to Trigonometric Form

The denominator is \(1+i\). Express it as a point in the complex plane: it has real part 1 and imaginary part 1. This corresponds to the point (1, 1).The modulus (or absolute value) is \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\).The angle \( \theta \) is 45° or \( \frac{\pi}{4} \) radians (as it lies along the line \(y=x\) in the first quadrant).Thus, the trigonometric form of \(1+i\) is \(\sqrt{2} \times (\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right))\).

Step 3: Apply the Quotient Formula for Complex Numbers

When dividing two complex numbers in trigonometric form: \[\frac{r_1(\cos \theta_1 + i\sin \theta_1)}{r_2(\cos \theta_2 + i\sin \theta_2)} = \frac{r_1}{r_2}\left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)\]For this problem:- \(r_1 = 1\), \(\theta_1 = \frac{3\pi}{2}\)- \(r_2 = \sqrt{2}\), \(\theta_2 = \frac{\pi}{4}\)Thus, we get:\[\frac{1}{\sqrt{2}}\left(\cos\left(\frac{3\pi}{2} - \frac{\pi}{4}\right) + i\sin\left(\frac{3\pi}{2} - \frac{\pi}{4}\right)\right)\]

Step 4: Calculate the Angles and Simplify

Calculate \(\theta_1 - \theta_2 = \frac{3\pi}{2} - \frac{\pi}{4} = \frac{6\pi}{4} - \frac{\pi}{4} = \frac{5\pi}{4}\).Since \(\theta_1 - \theta_2 = \frac{5\pi}{4}\) corresponds to 225° in standard position, we find:\(\cos\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}}\) and \(\sin\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}}\).Thus, the expression becomes:\[\frac{1}{\sqrt{2}}(-\frac{1}{\sqrt{2}} + i(-\frac{1}{\sqrt{2}})) = \frac{-1}{2} - i\frac{1}{2}\]

Step 5: Finalize in Rectangular Form

The final result in rectangular form is \(-\frac{1}{2} - i\frac{1}{2}\).Therefore, the quotient \(\frac{-i}{1+i}\) in rectangular form is \(-\frac{1}{2} - i\frac{1}{2}\).

Key Concepts

Rectangular Form

Complex numbers can be represented in different forms, and one of the most common is the rectangular form. In this form, a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part and \(b\) is the imaginary part of the complex number. For example, the complex number \(1 + i\) in rectangular form has a real part of 1 and an imaginary part of 1.
This method is very similar to representing a point on the Cartesian plane where the x-coordinate corresponds to the real part and the y-coordinate corresponds to the imaginary part. This makes it easy to visualize and perform arithmetic operations such as addition and subtraction.

• Helps in easy calculation of the sum and difference of complex numbers.
• Allows straightforward plot on the complex plane.

Understanding the rectangular form is crucial, as it is often the first step in analyzing complex numbers in any mathematical or engineering problem.

Trigonometric Form

When working with complex numbers, the trigonometric form can give more insight, especially when multiplying or dividing. This form expresses a complex number using its modulus (or absolute value) and an angle, denoted as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the modulus and \(\theta\) is the angle the complex number makes with the positive x-axis.
For instance, converting \(-i\) into trigonometric form involves finding its modulus, which is 1, and its angle, which is \(\frac{3\pi}{2}\). Consequently, \(-i\) in trigonometric form becomes \(1(\cos(\frac{3\pi}{2}) + i\sin(\frac{3\pi}{2}))\). This form is very useful for:

• Efficient handling of multiplication and division of complex numbers.
• Simplifying powers and roots of complex numbers using De Moivre's Theorem.

Thus, understanding and converting between these forms helps streamline complex operations, making them much more manageable.

Complex Plane

The complex plane is an invaluable tool for visualizing complex numbers. It allows a complex number to be represented as a point in a two-dimensional space, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.
In this plane, any complex number \(a + bi\) corresponds to the point \((a, b)\). For example, the point \(1 + i\) is located one unit along both the real (x-axis) and imaginary (y-axis) axes. This visualization aids greatly in understanding operations on complex numbers as geometrical transformations.

• Facilitates understanding of complex conjugates, with reflections across the real axis.
• Shows addition and subtraction as vector operations.

By using the complex plane, one can analyze and compute complex numbers more intuitively, translating abstract concepts into more tangible forms.

Complex Modulus

The complex modulus is the distance of a complex number from the origin in the complex plane. It is analogous to finding the magnitude of a vector. For a complex number \(a + bi\), the modulus is calculated using the formula \(|a + bi| = \sqrt{a^2 + b^2}\).
This modulus is essentially the length of the line segment from the origin to the point \((a, b)\) on the complex plane. In the example \(1 + i\), the modulus \(|1+i|\) is \(\sqrt{2}\), which indicates the distance of \(1+i\) from the origin is \(\sqrt{2}\).

• Provides a measure of the size of complex numbers, regardless of direction.
• Plays a key role in converting complex numbers between rectangular and trigonometric forms.

Knowing how to compute and interpret the modulus is vital, especially in complex arithmetic, as it provides insights into the magnitude and "size" of the complex number.