Problem 43 - College Trigonometry

Question

Use \(z=-\frac{3 \sqrt{3}}{2}+\frac{3}{2} i\) and \(w=3 \sqrt{2}-3 i \sqrt{2}\) to compute the quantity. Express your answers in polar form using the principal argument. $$ \frac{w}{z} $$

Step-by-Step Solution

Answer

\(\frac{w}{z} = 2(\cos \frac{11\pi}{12} + i \sin \frac{11\pi}{12})\)

Step 1: Express Each Complex Number in Rectangular Form

First, make sure both complex numbers \(z\) and \(w\) are correctly identified in rectangular form. \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\) is already in rectangular form. \(w = 3 \sqrt{2} - 3 \sqrt{2} i\) is also in rectangular form.

Step 2: Find the Modulus of Each Complex Number

The modulus of a complex number \(a + bi\) is given by \(|a + bi| = \sqrt{a^2 + b^2}\). For \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\), the modulus is:\[|z| = \sqrt{ \left(-\frac{3 \sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2 } = \sqrt{\frac{27}{4} + \frac{9}{4}} = \sqrt{\frac{36}{4}} = 3.\]For \(w = 3 \sqrt{2} - 3 \sqrt{2} i\), the modulus is:\[|w| = \sqrt{(3\sqrt{2})^2 + (-3\sqrt{2})^2} = \sqrt{18 + 18} = \sqrt{36} = 6.\]

Step 3: Find the Argument of Each Complex Number

The argument \(\theta\) of a complex number \(a + bi\) is given by \(\tan^{-1}\left(\frac{b}{a}\right)\), adjusted for the quadrant. For \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\), we have:\[\text{Argument of } z = \tan^{-1} \left( \frac{\frac{3}{2}}{-\frac{3\sqrt{3}}{2}} \right) = \tan^{-1} \left( -\frac{1}{\sqrt{3}} \right),\]which is equivalent to \(-\frac{\pi}{6}\) in the fourth quadrant. Therefore, the principal argument is \(\theta = \frac{5\pi}{6}\).For \(w = 3 \sqrt{2} - 3 \sqrt{2} i\), the argument is:\[\text{Argument of } w = \tan^{-1}\left( \frac{-3\sqrt{2}}{3\sqrt{2}} \right) = \tan^{-1}(-1) = -\frac{\pi}{4}.\] Adjust for the fourth quadrant, which places it at \(-\frac{\pi}{4}\) or equivalently \(\frac{7\pi}{4}\) as the principal value.

Step 4: Express Each Complex Number in Polar Form

The polar form of a complex number is \(r(\cos \theta + i\sin \theta)\) or \(re^{i\theta}\). For \(z\), use:\[z = 3 \left(\cos \frac{5\pi}{6} + i\sin \frac{5\pi}{6} \right)\]For \(w\), use:\[w = 6 \left( \cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right)\]

Step 5: Compute the Quotient \(\frac{w}{z}\) in Polar Form

When dividing complex numbers in polar form \(\frac{w}{z} = \frac{r_w}{r_z}\left(\cos(\theta_w - \theta_z) + i\sin(\theta_w - \theta_z)\right)\). The moduli divide as follows:\[\frac{|w|}{|z|} = \frac{6}{3} = 2\]The arguments subtract:\[\theta_W - \theta_Z = \frac{7\pi}{4} - \frac{5\pi}{6} = \frac{21\pi}{12} - \frac{10\pi}{12} = \frac{11\pi}{12}\]So, the quotient in polar form is:\[\frac{w}{z} = 2 \left( \cos \frac{11\pi}{12} + i \sin \frac{11\pi}{12} \right)\]

Key Concepts

Polar Form

Complex numbers can be expressed in polar form, which highlights their modulus and argument. Unlike the rectangular form where numbers are expressed as \(a + bi\), polar form represents complex numbers as \(r(\cos \theta + i \sin \theta)\) or simply \(re^{i\theta}\). Here, \(r\) represents the modulus of the complex number, while \(\theta\) stands for the argument.Expressing numbers in polar form helps simplify operations like multiplication and division. This is because multiplication involves adding angles and multiplying moduli, while division involves subtracting angles and dividing moduli. This makes understanding operations on complex numbers much more intuitive.

Modulus of a Complex Number

The modulus of a complex number is like the number's "distance" from the origin on the complex plane. For a complex number \(a + bi\), the modulus is calculated using the formula:

• \(|a + bi| = \sqrt{a^2 + b^2}\)

It gives a real number as the output, showing how far the number is from the origin. In our exercise, for \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\), we computed the modulus to be 3, and for \(w = 3 \sqrt{2} - 3i \sqrt{2}\), it was found to be 6. Knowing the modulus is an important step in converting a complex number into polar form.

Argument of a Complex Number

The argument of a complex number is the angle that it forms with the positive real axis, measured counterclockwise. It's often determined using the arctan function. For a complex number \(a + bi\), the argument \(\theta\) can be found using:

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

However, it's crucial to consider the quadrant in which the complex number is located to get the principal argument.For \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\), the argument was calculated to be \(\frac{5\pi}{6}\). For \(w = 3\sqrt{2} - 3\sqrt{2} i\), the argument is \(\frac{7\pi}{4}\). These angles are vital when you express complex numbers in polar form.

Rectangular Form

The rectangular form of a complex number is the most straightforward way to express it, and is written as \(a + bi\). Here, \(a\) is the real part, while \(b\) is the imaginary part. This form makes it easy to perform additions and subtractions, as you just need to group like terms together.In this exercise, both \(z\) and \(w\) were already given in rectangular form. Understanding this form is important as it's often the starting point for converting to other representations, such as polar form.

Division of Complex Numbers

Dividing complex numbers can be a bit tricky in rectangular form, but in polar form, it's quite elegant. Division involves subtracting the arguments and dividing the moduli of the numbers. If \(w\) and \(z\) are in polar form, say \(w = r_w e^{i\theta_w}\) and \(z = r_z e^{i\theta_z}\), their division is:

• \(\frac{w}{z} = \frac{r_w}{r_z}e^{i(\theta_w - \theta_z)}\)

Here, the modulus of the quotient is \(\frac{r_w}{r_z}\) and the argument is simply \(\theta_w - \theta_z\).For this exercise, the final quotient \(\frac{w}{z}\) was \(2(\cos \frac{11\pi}{12} + i \sin \frac{11\pi}{12})\). This gives you an efficient way to handle division, while adhering to the elegance of unity found in polar representation.