Question

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\frac{1}{2}\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)\right]^{5} $$

Short answer

Rectangular form: \[\frac{1}{32} + 0i\]. Exponential form: \[\frac{1}{32}\].

Step-by-step solution

- Identify the given expression in polar form

The given expression is \[\frac{1}{2} \left(\cos \frac{2 \pi}{5} + i \sin \frac{2 \pi}{5}\right)\]. This is in the form \[r \cdot \text{cis} \theta\] where \[r = \frac{1}{2}\] and \[\theta = \frac{2 \pi}{5}\].

- Convert to exponential form

In exponential form, \[\text{cis} \theta = e^{i \theta}\]. Thus, \[\frac{1}{2} \left(\cos \frac{2 \pi}{5} + i \sin \frac{2 \pi}{5}\right) = \frac{1}{2} e^{i \frac{2\pi}{5}}\].

- Raise the expression to the power of 5

Using the property of exponents \[(re^{i\theta})^n = r^n e^{i n \theta}\], we have: \[\left(\frac{1}{2} e^{i \frac{2\pi}{5}}\right)^5 = \left(\frac{1}{2}\right)^5 e^{i 5 \cdot \frac{2\pi}{5}} = \frac{1}{32} e^{i 2\pi}\].

- Simplify the exponential expression

\[e^{i 2\pi} = \cos 2\pi + i \sin 2\pi = 1\]. Therefore, the expression simplifies to: \[\frac{1}{32} \cdot 1 = \frac{1}{32}\].

- Write the final rectangular and exponential forms

The rectangular form is: \[\frac{1}{32} + 0i\]. The exponential form is: \[\frac{1}{32} e^{i 0}\] or simply \[\frac{1}{32}\], since \[e^{i0} = 1\].

Key concepts

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit \(i\) is defined as the square root of \(-1\). This means \(i^2 = -1\).
Complex numbers are very useful in various branches of mathematics and engineering because they allow us to represent and solve equations that cannot be solved using only real numbers. Understanding complex numbers is crucial for converting between different forms, like rectangular, polar, and exponential.

exponential form

The exponential form of a complex number is a compact and elegant way to express complex numbers using the exponential function and Euler's formula.
Eulr's formula states that \(e^{i\theta} = \cos \theta + i \sin \theta\).
So, any complex number in polar form \(r \cdot (\cos \theta + i \sin \theta)\) can also be written in exponential form as \(r \cdot e^{i\theta}\).
Converting complex numbers to exponential form is especially handy when we need to perform operations like multiplication, division, and finding powers.

rectangular form

The rectangular form of a complex number is perhaps the most familiar representation.
It is written as \(a + bi\), where \(a\) stands for the real part and \(b\) stands for the coefficient of the imaginary part.
For example, in the solution, we found the rectangular form of the given problem to be \(\frac{1}{32} + 0i \). This clearly shows that the real part is \(\frac{1}{32}\) and the imaginary part is \(0\).
Rectangular form is straightforward to understand and excellent for addition and subtraction of complex numbers.

polar form

Polar form of a complex number uses the magnitude \(r\) and angle \(\theta\) to express the number.
It is written as \(r \cdot (\cos \theta + i \sin \theta)\).
This form is particularly useful for analyzing complex numbers in the context of rotation and scaling transformations.
In the given problem, the number was initially in polar form as \(\frac{1}{2} \( \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5} \)\).
Polar form makes it very easy to multiply and divide complex numbers by manipulating the magnitudes and adding or subtracting the angles.

Euler's formula

Euler's formula is a fundamental equation in complex analysis.
It links complex exponentials to trigonometric functions by stating \(e^{i\theta} = \cos \theta + i \sin \theta\).
This is invaluable in the conversion process between polar and exponential forms.
In the problem at hand, we used Euler’s formula to convert \(\frac{1}{2} \( \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5} \)\) to its exponential form \(\frac{1}{2} e^{i \frac{2\pi}{5}}\).
Euler's formula succinctly captures the essential relationship between trigonometry and exponential growth, making it a vital tool in complex number analysis and transformations.