Problem 72 - 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. $$\frac{2 i}{-1-i \sqrt{3}}$$

Step-by-Step Solution

Answer

The quotient in rectangular form is \(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\).

Step 1: Express the Numerator in Trigonometric Form

The numerator is already in the form of a pure imaginary number: \(2i\). We identify the modulus as \(|2i| = 2\) and the argument as \(\frac{\pi}{2}\) since it is along the positive imaginary axis. The trigonometric form of the numerator is thus \(2(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\).

Step 2: Express the Denominator in Trigonometric Form

For the denominator \(-1 - i \sqrt{3}\), calculate the modulus: \(\sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{4} = 2\). The argument is determined by the angle in the fourth quadrant, so \(\tan^{-1}\Bigg(\frac{-\sqrt{3}}{-1}\Bigg) = \frac{5\pi}{3}\), giving the trigonometric form: \(2(\cos \frac{5\pi}{3} + i\sin \frac{5\pi}{3})\).

Step 3: Divide the Complex Numbers in Trigonometric Form

Divide the moduli and subtract the arguments. The magnitude of the quotient is \(\frac{2}{2} = 1\). The argument of the quotient is \(\frac{\pi}{2} - \frac{5\pi}{3} = \frac{-7\pi}{6}\). In standard form, this is equivalent to \(\frac{5\pi}{6}\). Thus, the trigonometric form of the quotient is \(\cos \frac{5\pi}{6} + i\sin \frac{5\pi}{6}\).

Step 4: Convert Back to Rectangular Form

Evaluate \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\) and \(\sin \frac{5\pi}{6} = \frac{1}{2}\). Therefore, the rectangular form of the quotient is \(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\).

Key Concepts

Trigonometric Form

Complex numbers can be represented in trigonometric form, which can be particularly useful for division and multiplication. This form utilizes a number's modulus and argument. The modulus is simply the distance from the origin to the point in the complex plane, and it is denoted as \(|z|\). The argument, or angle \(\theta\), is the angle formed with the positive x-axis. Together, these allow the expression of a complex number \(z = a + bi\) in the form \(|z|(\cos \theta + i\sin \theta)\). This form is beneficial because it simplifies multiplying and dividing complex numbers.

Rectangular Form

The rectangular form, also known as the Cartesian form, is a way to express complex numbers using real and imaginary parts, written as \(a + bi\). Here, \(a\) is the real part, and \(b\) is the imaginary part of the complex number. It is called rectangular form because it corresponds to a point \((a, b)\) in the coordinate plane. This form is often more intuitive for addition and subtraction of complex numbers but can be less convenient for division, especially when converting from trigonometric forms.

Complex Division

Dividing complex numbers can be simplified by using their trigonometric forms. The rule is that you divide the moduli and subtract the arguments. For two complex numbers \(z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\), their quotient \(z = \frac{z_1}{z_2}\) is calculated as \(\frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))\). This method greatly simplifies the arithmetic involved compared to dividing in rectangular form, which usually requires rationalizing the denominator.