Problem 46 - Algebra and Trigonometry

Question

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[3\left(\cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9}\right)\right]^{3} $$

Step-by-Step Solution

Answer

Rectangular form: \(-\frac{27}{2} - i\frac{27\sqrt{3}}{2}\). Exponential form: \(27 e^{i \frac{4 \pi}{3}}\).

Step 1: Expand the given expression

The given expression is \ \ \( \left[3\left(\cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9}\right)\right]^{3} \). We will use the fact that \( re^{i\theta} = r (\cos \theta + i \sin \theta) \). Let's rewrite it in exponential form first: \ \ \(3e^{i \frac{4 \pi}{9}} \).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that \( (re^{i\theta})^n = r^n e^{i n \theta} \). Here, \( r = 3 \) and \( \theta = \frac{4 \pi}{9} \), and we need to raise the expression to the power of 3. \ \ So, \( [3 e^{i \frac{4 \pi}{9}}]^3 = 3^3 e^{i * 3 * \frac{4 \pi}{9}} \). Calculate each term: \ \ \( 3^3 = 27 \) and \( 3 \times \frac{4 \pi}{9} = \frac{12 \pi}{9} = \frac{4 \pi}{3} \). Thus we get: \ \ \( 27 e^{i \frac{4 \pi}{3}} \).

Step 3: Convert to rectangular form

Now, to convert from exponential form to rectangular form, recall that \( re^{i\theta} = r(\cos \theta + i \sin \theta) \). Here, \( r = 27 \) and \( \theta = \frac{4 \pi}{3} \). Therefore, we have: \ \ \( 27(\cos \frac{4 \pi}{3} + i \sin \frac{4 \pi}{3}) \). \ \ Using trigonometric values, \( \cos \frac{4 \pi}{3} = -\frac{1}{2} \) and \( \sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2} \). Substituting these values gives: \ \ \( 27(-\frac{1}{2} + i(-\frac{\sqrt{3}}{2})) = 27(-\frac{1}{2} - i\frac{\sqrt{3}}{2}) = -\frac{27}{2} - i\frac{27\sqrt{3}}{2} \).

Key Concepts

rectangular form

To fully understand complex numbers, it's vital to recognize their different forms. The rectangular form of a complex number is a way to represent it using real and imaginary components. This form is written as \( x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. For instance, in the problem, we converted the expression into rectangular form as \(-\frac{27}{2} - i\frac{27\sqrt{3}}{2}\). This expression indicates that the real part is \(-\frac{27}{2}\) and the imaginary part is \(-\frac{27\sqrt{3}}{2}\). Understanding this helps to plot complex numbers on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This visualization can make solving more complex problems easier by conceptualizing their position and behavior.

exponential form

Complex numbers can also be represented using the exponential form. It's derived from Euler's formula: \( re^{i\theta} = r (\cos \theta + i \sin \theta) \).Here, \( r \) denotes the magnitude, or modulus, of the complex number, and \( \theta \) represents the argument, or angle. In the problem, the exponential form of the expression was \( 27e^{i \frac{4\pi}{3}} \). Exponential form is particularly useful in multiplication and division of complex numbers, as it simplifies calculations involving powers and roots. By converting a complex number from rectangular form to exponential form, we significantly simplify the exponentiation and extraction of roots.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool when dealing with powers and roots of complex numbers. The theorem is stated as \( (re^{i\theta})^n = r^n e^{i n \theta} \).In our example, we used De Moivre's Theorem to raise the expression to the power of 3. We had \( 3e^{i \frac{4\pi}{9}} \) raised to the third power, resulting in \( 27e^{i \frac{4\pi}{3}} \). De Moivre’s Theorem thus allowed us to handle complex number exponentiation easily by dealing separately with the magnitude \( r \) and the angle \( \theta \). It essentially breaks down a complex exponentiation problem into simpler multiplicative steps.

trigonometric values

Trigonometric values are crucial when converting between the exponential form and rectangular form of complex numbers. For angles like \( \frac{4\pi}{3} \), it helps to know the trigonometric values, \( \cos \frac{4\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2} \). By substituting these values back into the exponential form, \( re^{i\theta} = r (\cos \theta + i \sin \theta) \), we were able to convert \( 27e^{i \frac{4\pi}{3}} \) to \( -\frac{27}{2} - i\frac{27\sqrt{3}}{2} \). Recognizing and recalling these trigonometric values for standard angles simplifies converting complex numbers between different forms.

algebraic manipulation

Algebraic manipulation plays a significant role when dealing with complex numbers. It helps simplify expressions and solve equations involving complex numbers. In the given problem, we used algebraic manipulation to apply De Moivre's Theorem by breaking down the exponent into simpler terms. For instance, after applying the theorem, we calculated \( 3^3 \) as 27, and \( 3 \times \frac{4\pi}{9} \) as \( \frac{12\pi}{9} = \frac{4\pi}{3} \). This resulted in a much simpler expression \( 27e^{i \frac{4\pi}{3}} \). Finally, we substituted the trigonometric values to convert back to rectangular form. Efficient algebraic manipulation reduces computation complexity and aids in effective problem-solving.