Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient in rectangular form, using exact values. $$\frac{4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)}{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}$$

Step-by-Step Solution

Answer

The quotient is \(\sqrt{3} - i\).

Step 1: Convert to Rectangular Form

First, let's convert the complex numbers from polar form to rectangular form. Using the identity \(a + bi = r(\cos\theta + i\sin\theta)\), we find:For the numerator: \(4(\cos 120^{\circ} + i \sin 120^{\circ})\)- \(\cos 120^{\circ} = -\frac{1}{2}\)- \(\sin 120^{\circ} = \frac{\sqrt{3}}{2}\) This gives us:\[4 \left( -\frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = -2 + 2i\sqrt{3}\]For the denominator: \(2(\cos 150^{\circ} + i \sin 150^{\circ})\)- \(\cos 150^{\circ} = -\frac{\sqrt{3}}{2}\)- \(\sin 150^{\circ} = \frac{1}{2}\) This gives us:\[2 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = -\sqrt{3} + i\]

Step 2: Write the Division of Complex Numbers

Now express the division of complex numbers in rectangular form:\[\frac{-2 + 2i\sqrt{3}}{-\sqrt{3} + i}\]

Step 3: Multiply by the Conjugate

To simplify the division, multiply the numerator and denominator by the conjugate of the denominator.The conjugate of \(-\sqrt{3} + i\) is \(-\sqrt{3} - i\).Multiply:\[\left( \frac{-2 + 2i\sqrt{3}}{-\sqrt{3} + i} \right) \times \left( \frac{-\sqrt{3} - i}{-\sqrt{3} - i} \right) = \frac{(-2 + 2i\sqrt{3})(-\sqrt{3} - i)}{(-\sqrt{3} + i)(-\sqrt{3} - i)}\]

Step 4: Simplify the Denominator

First, simplify the denominator:\[(-\sqrt{3} + i)(-\sqrt{3} - i) = (-\sqrt{3})^2 - (i)^2 = 3 - (-1) = 4\]

Step 5: Distribute in the Numerator

Now expand the numerator:\[-2(-\sqrt{3}) + (-2)(-i)\]\[+ 2i\sqrt{3}(-\sqrt{3}) + 2i\sqrt{3}(-i)\]Calculates to:\[2\sqrt{3} + 2i - 6i - 2\sqrt{3}i^2\]Since \(i^2 = -1\), then:\[2\sqrt{3} + 2i - 6i + 2\sqrt{3} = 4\sqrt{3} - 4i\]

Step 6: Final Calculation

Calculate:\[\frac{4\sqrt{3} - 4i}{4} = \sqrt{3} - i\]

Step 7: Write the Final Answer

Therefore, the quotient in rectangular form is \(\sqrt{3} - i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are typically expressed in the form \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component. The unit \(i\) represents the square root of \(-1\). By using these components, complex numbers can represent quantities that cannot be captured by purely real numbers.

• The standard form: \(a + bi\)
• The imaginary unit \(i\): \(i^2 = -1\)
• Complex numbers can be graphed on the complex plane, where the x-axis represents the real part, and the y-axis represents the imaginary part.

Complex numbers have important applications in various fields like engineering, physics, and mathematics, as they provide more robust solutions to problems involving roots of equations and oscillatory behaviors.

Quotient of Complex Numbers

The quotient of complex numbers is akin to division in the realm of complex numbers. It involves dividing one complex number by another. However, because complex numbers have both real and imaginary components, division is slightly more complex than that of real numbers.
To divide two complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. This method eliminates the imaginary part in the denominator, simplifying the complex division.

• Write the division in rectangular form: If you have two complex numbers, \( rac{a + bi}{c + di} \).
• Multiply by the conjugate of the denominator: \( rac{c - di}{c - di} \).
• Simplify the expression to obtain a single complex number, typically in the form of \( x + yi \).

Operationally, the trick is to convert the division into multiplication, making use of the conjugate to simplify.

Rectangular and Polar Form Conversion

Complex numbers can also be represented in polar form, which provides a different perspective using the concepts of magnitude and direction. This is particularly useful for operations like multiplication and division.

• Rectangular form: \( a + bi \)
• Polar form: \( r(\cos \theta + i \sin \theta) \)

The conversion between these forms relies on trigonometric identities.
- To convert from polar to rectangular, you use:

• Real part: \( a = r \cos \theta \)
• Imaginary part: \( b = r \sin \theta \)

- To go from rectangular to polar, calculate:

• Magnitude: \( r = \sqrt{a^2 + b^2} \)
• Angle: \( \theta = \tan^{-1}(b/a) \)

Understanding both forms and their interchangeability is crucial for fully mastering complex arithmetic.

Conjugate of Complex Numbers

The conjugate of a complex number is simply the number with the imaginary part sign reversed. If you have a complex number in the form \( a + bi \), its conjugate is \( a - bi \). The conjugate is a cornerstone concept when dealing with complex number operations, particularly division, because it allows us to "clear" the imaginary part from the denominator.

• Conjugate pairs: \( (a + bi, a - bi) \)
• The product of a complex number and its conjugate is a real number: \( (a + bi)(a - bi) = a^2 + b^2 \)

Using conjugates effectively simplifies division and other operations by eliminating the imaginary parts when needed. The simplicity it provides in the arithmetic of complex numbers plays a crucial role in their practical computation and application.