Problem 17 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=9\left[\cos \left(\frac{5 \pi}{12}\right)+i \sin \left(\frac{5 \pi}{12}\right)\right] \text { and } z_{2}=3\left[\cos \left(\frac{\pi}{12}\right)+i \sin \left(\frac{\pi}{12}\right)\right]$$

Step-by-Step Solution

Answer

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

Step 1: Identify Given Complex Numbers

The given complex numbers are \(z_1 = 9 \left[ \cos \left( \frac{5 \pi}{12} \right) + i \sin \left( \frac{5 \pi}{12} \right) \right]\) and \(z_2 = 3 \left[ \cos \left( \frac{\pi}{12} \right) + i \sin \left( \frac{\pi}{12} \right) \right]\). Both are provided in polar form: \(z = r (\cos \theta + i \sin \theta)\).

Step 2: Apply Division Formula for Complex Numbers

To divide two complex numbers in polar form \(z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)\), use the formula: \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right)\).

Step 3: Calculate Magnitude of Quotient

Calculate the magnitude of the quotient: \(\frac{9}{3} = 3\).

Step 4: Determine Angle of the Quotient

Subtract the angles: \(\theta_1 = \frac{5\pi}{12}\) and \(\theta_2 = \frac{\pi}{12}\). Using \(\theta_1 - \theta_2 = \frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}\).

Step 5: Express Quotient in Polar Form

The quotient in polar form is \(3 \left( \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) \right)\).

Step 6: Convert to Rectangular Form

We know \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\). Thus, \(3 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = \frac{3}{2} + i\frac{3\sqrt{3}}{2}\).

Key Concepts

Polar Form

When dealing with complex numbers, the polar form is a powerful tool. It represents complex numbers in terms of their modulus and argument, typically written as \( z = r ( \cos \theta + i \sin \theta ) \). Here, \(r\) denotes the modulus or magnitude of the complex number, while \(\theta\) is the argument or angle. This form is particularly beneficial for multiplication and division of complex numbers, as illustrated in our example.

• The modulus \(r\) is the distance from the origin to the point in the complex plane, making it a measure of the size of the number.
• The argument \(\theta\) is the angle formed with the positive x-axis, indicating the direction of the number.

Converting a complex number to polar form involves finding these two components. In the original exercise, both \(z_1\) and \(z_2\) are presented in polar form, making it straightforward to apply formulas like the division formula used here.

Rectangular Form

The rectangular form of a complex number expresses it as \( a + bi \), where \(a\) is the real part and \(b\) is the imaginary part. For some, this form is more intuitive since it aligns with standard Cartesian coordinates.

• In rectangular form, complex numbers appear identical to ordered pairs \((a, b)\) on the complex plane.
• It captures both magnitude and direction but through real and imaginary parts instead of polar coordinates.

Converting a polar form back to rectangular form involves using trigonometric identities. In the exercise, after finding the polar form of the quotient, it was converted to rectangular form using \( \cos(\theta) \) and \( \sin(\theta) \) of the angle found. The results are multiplied by the magnitude to get the final form: \( \frac{3}{2} + i\frac{3\sqrt{3}}{2} \).

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers, often confused with polar form, is expressed as \( z = r ( \cos \theta + i \sin \theta ) \), much like the polar form. This expression highlights the trigonometric representation of angles and can be a helpful bridge between polar and rectangular forms.

• Using the trigonometric functions, we describe both the orientation and length of the vector representing the complex number.
• It emphasizes the use of sine and cosine to establish the direction, leveraging their periodic nature.

The trigonometric form is particularly advantageous for understanding operations like multiplication and division because angles can be easily added or subtracted. In the problem, the division of the complex numbers was simplified using the subtraction of angles, which exemplifies this utility. This form underscores how trigonometry aids in complex number calculations.