Problem 49 - Algebra and Trigonometry

Question

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\sqrt{3}\left(\cos \frac{\pi}{18}+i \sin \frac{\pi}{18}\right)\right]^{6} $$

Step-by-Step Solution

Answer

Rectangular form: 13.5 + 13.5i\sqrt{3}Exponential form: 27e^{i \frac{\pi}{3}}

Step 1 - Identify the given form

The expression \[\[\begin{equation}\begin{split}\left[\sqrt{3} \left( \cos \frac{\pi}{18} + i \sin \frac{\pi}{18} \right)\right]^{6}\end{split}\end{equation}\]\] is in the polar form where \[ r = \sqrt{3} \quad \text{and} \quad \theta = \frac{\pi}{18}\].

Step 2 - Apply De Moivre's Theorem

According to De Moivre's Theorem, for a complex number in polar form \[r( \cos \theta + i \sin \theta)\] raised to a power \[n\], we have:\[\left[r \left( \cos \theta + i \sin \theta \right) \right]^{n} = r^{n} \left( \cos(n \theta) + i \sin(n \theta) \right)\]. Apply this theorem to the expression.

Step 3 - Calculate the magnitude

Calculate \[ r^6 \]. Since \[ r = \sqrt{3} \], we have \[( \sqrt{3})^6 = 3^3 = 27 \].

Step 4 - Calculate the angle

Calculate \[ 6 \theta \]. Since \[ \theta = \frac{\pi}{18} \], \[ 6 \theta = 6 \cdot \frac{\pi}{18} = \frac{6\pi}{18} = \frac{\pi}{3} \].

Step 5 - Write in polar form

Therefore, the expression in polar form is: \[ 27 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \].

Step 6 - Write in rectangular form

Evaluate \[ \cos \frac{\pi}{3} \] and \[ \sin \frac{\pi}{3} \]. Since \[ \cos \frac{\pi}{3} = \frac{1}{2} \] and \[ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \], we have:\[ 27 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = 27 \cdot \frac{1}{2} + 27 \cdot i \frac{\sqrt{3}}{2} = 13.5 + i 13.5\sqrt{3} \].

Step 7 - Write in exponential form

The exponential form of the expression can be written as: \[ r e^{i \theta} \]. Using \[ r = 27 \] and \[ \theta = \frac{\pi}{3} \], we get: \[ 27 e^{i \frac{\pi}{3}} \].

Key Concepts

Polar Form

A complex number in polar form is represented as \[ r (\cos \theta + i \sin \theta) \]. In this representation, \[ r \] is the magnitude (or modulus) of the complex number, and \[ \theta \] is the angle (or argument) with the positive real axis measured counterclockwise. To convert from rectangular form to polar form, we use the formulas: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \tan^{-1}(\frac{y}{x}) \]. This form is particularly useful for multiplying and dividing complex numbers, as well as for raising them to powers.

Rectangular Form

The rectangular (or Cartesian) form of a complex number is given by \[ x + yi \], where \[ x \] is the real part and \[ y \] is the imaginary part. For instance, in our example, we found the rectangular form of the complex number to be \[ 13.5 + 13.5 \sqrt{3} i \]. This form is often the most intuitive for representing and visualizing complex numbers. When converting from polar form to rectangular form, you simply compute \[ x = r \cos \theta \] and \[ y = r \sin \theta \].

De Moivre's Theorem

De Moivre's Theorem provides a powerful way to raise complex numbers in polar form to a power. It states: \[ \left[r( \cos \theta + i \sin \theta) \right]^n = r^n ( \cos(n \theta) + i \sin(n \theta)) \]. In our example, we used this theorem to simplify \[ \left[ \sqrt{3}( \cos \frac{\pi}{18} + i \sin \frac{\pi}{18}) \right]^6 \]. By applying this theorem, we calculated \[ 27 ( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \], easing us into converting from polar to both rectangular and exponential forms efficiently.

Exponential Form

The exponential form of a complex number is another powerful representation and is written as \[ r e^{i \theta} \]. This form utilizes Euler's formula: \[ e^{i \theta} = \cos \theta + i \sin \theta \]. This makes converting between polar and exponential forms straightforward. For the given problem, the exponential form was found to be \[ 27 e^{i \frac{\pi}{3}} \]. This form is particularly useful for simplifying multiplication and division of complex numbers, as well as for deeper understanding in fields like electrical engineering and quantum physics.