Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\frac{16\left(\cos 300^{\circ}+i \sin 300^{\circ}\right)}{8\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)}$$

Step-by-Step Solution

Answer

The product in rectangular form is \(-1 - i\sqrt{3}\).

Step 1: Simplify the Coefficients

First, simplify the fraction of the coefficients outside the trigonometric terms. The expression is \[ \frac{16}{8} \left( \cos 300^{\circ} + i \sin 300^{\circ} \right) \div \left( \cos 60^{\circ} + i \sin 60^{\circ} \right). \]Simplifying \( \frac{16}{8} \) gives us 2.

Step 2: Use Trigonometric Identities

Next, we express the complex numbers in terms of Euler's formula: \[ 2 \cdot \left( \cos 300^{\circ} + i \sin 300^{\circ} \right) \cdot \frac{1}{\cos 60^{\circ} + i \sin 60^{\circ}}. \]

Step 3: Convert Angles to Standard Form

Convert angles from degrees to rectangular form using known trigonometric identities:- \(\cos 300^{\circ} = \frac{1}{2}\) and \(\sin 300^{\circ} = -\frac{\sqrt{3}}{2}\).- \(\cos 60^{\circ} = \frac{1}{2}\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).

Step 4: Simplify Using Rectangular Form

Substitute these values into the expression:\[ 2 \cdot \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right) \div \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right). \]

Step 5: Multiply by the Conjugate

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator:The conjugate of \( \frac{1}{2} + i \frac{\sqrt{3}}{2} \) is \( \frac{1}{2} - i \frac{\sqrt{3}}{2} \). Multiply both the numerator and the denominator by this:\[ \frac{ \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right)^2}{\left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right)}. \]

Step 6: Simplify the Denominator

Calculate the denominator using the difference of squares:\[ \left( \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{1}{4} + \frac{3}{4} = 1. \]So, the denominator simplifies to 1.

Step 7: Simplify the Numerator

Expand the square in the numerator:\[ \left( \frac{1}{2} - i \frac{\sqrt{3}}{2} \right)^2 = \left( \frac{1}{4} - i \frac{\sqrt{3}}{2} \cdot \frac{1}{2} - i \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + i^2 \frac{3}{4} \right). \]This simplifies to:\[ \frac{1}{4} - i\sqrt{3} \cdot \frac{1}{2} + \frac{3}{4}(-1) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}. \]

Step 8: Final Result

Multiply the result of Step 7 by 2 (from Step 1):\[ 2 \cdot \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = -1 - i\sqrt{3}. \]This is the product in rectangular form.

Key Concepts

Euler's Formula

Euler's Formula is a fundamental bridge between trigonometry and complex numbers. The formula is given by \( e^{i\theta} = ext{cos}\theta + i\text{sin}\theta \). This simple equation beautifully unites exponential growth with circular motion, explaining how complex exponentials correspond to rotations in the complex plane.

• **Exponential Form:** Moving beyond the real numbers, Euler's formula allows us to express complex numbers in polar form using exponential notation. This is compact and often simplifies multiplication and division.
• **Understanding Rotations:** If we think of \(\theta \) as an angle, and the expression \( e^{i\theta} \) represents a point on the unit circle, this point rotates counterclockwise as \(\theta\) increases. This concept is especially useful when solving problems involving trigonometry in the complex plane.

In the given problem, Euler’s formula helps to transform the expressions \( ext{cos}\theta + i\text{sin}\theta \) into simpler calculations during multiplication or division.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and lengths of triangles. They are fundamental in simplifying expressions involving trigonometric functions.

• **Common Identities:** Use identities like \( \cos30^{\circ} = \frac{1}{2} \) and \( \sin30^{\circ} = \frac{\sqrt{3}}{2} \) to convert angles in trigonometric terms to standard numerical values. This step often simplifies complex problems by transforming trigonometric expressions into basic arithmetic forms.
• **Angle Modifications:** Trigonometric identities help convert complex numbers given in degrees to equivalent forms that are easier to handle algebraically. For example, calculations involving \( \cos300^{\circ} \) leverage the known cosine values of 60-degree angles.

In our problem, substituting these identities into the expression allowed both the numerator and denominator to be translated into familiar forms for further simplification.

Rectangular Form

The rectangular form of a complex number expresses it in the format \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This representation is particularly useful for addition and subtraction of complex numbers.

• **Representation:** When a complex number is given in terms of sines and cosines, such as \( \text{cos}\theta + i\text{sin}\theta \), transforming it into \( a + bi \) makes it easier to perform arithmetic operations.
• **Simplification Process:** In our exercise, converting trigonometric expressions like \( \text{cos}300^{\circ} + i\text{sin}300^{\circ} \) into \( \frac{1}{2} - i\frac{\sqrt{3}}{2} \) and similarly for \( \text{cos}60^{\circ} + i\text{sin}60^{\circ} \) allows for straightforward application of algebraic manipulations.

Understanding the rectangular form is key to simplifying trigonometric-based expressions and moving towards the final solution.

Conjugate in Complex Numbers

The conjugate of a complex number is a fundamental tool, especially in division. For a complex number \( a + bi \), its conjugate is \( a - bi \). This simple change plays a crucial role in eliminating the imaginary part from the denominator.

• **Rationalization of Denominators:** Multiplying both numerator and denominator by the conjugate of the denominator turns the denominator into a real number \( (a^2 + b^2) \). This eliminates the imaginary component, simplifying computations.
• **Simplification and Stability:** In division, using conjugates helps simplify the calculation by reducing it to operations with real numbers, which are easier to manage and less prone to computational errors.

The exercise showcases this technique when the expression is multiplied by the conjugate of \( \frac{1}{2} + i\frac{\sqrt{3}}{2} \), appropriately working through complexity to simplify the problem into a manageable format.