Question

Write the following in rectangular form, $z=a+ib$. a. $4e^{i\pi /6}$ b. $\sqrt{2}{e}^{5i\pi /4}$ c. $(1-i{)}^{100}.$

Short answer

a. $2\sqrt{3}+2i$, b. $-1+i$, c. $-1$

Step-by-step solution

Problem a: Convert $4e^{i\pi /6}$ to Rectangular Form

From Euler's formula, this can be written as $4(\cos (\pi /6)+i\sin (\pi /6))$. Then, using the values cos($\pi /6$ = $\sqrt{3}/2$, and \sin(\pi / 6) = 0.5, simplification gives $2\sqrt{3}+2i$.

Problem b: Convert $\sqrt{2}{e}^{5i\pi /4}$ to Rectangular Form

Using Euler's formula, this can be written as $\sqrt{2}(\cos (5\pi /4)+i\sin (5\pi /4))$. Knowing that \cos($5\pi /4$) = -$\frac{1}{\sqrt{2}}$, and \sin($5\pi /4$) = $\frac{1}{\sqrt{2}}$, simplification gives $-1+i$.

Problem c: Convert $(1-i{)}^{100}$ to Rectangular Form

The first step is to write $1-i$ in polar form. This gives $e^{7i\pi /4}$, due to $\cos (7\pi /4)=\sqrt{2}/2$ and $\sin (7\pi /4)=-\sqrt{2}/2$. Raising this to the power 100 gives $e^{175i\pi }=(e^{i\pi }{)}^{175}$.\nThe 175 is an odd number, and we know that $e^{i\pi }=-1$, so it can be simplified as $-1$.

Key concepts

Euler's Formula

Euler's Formula is a powerful tool in complex number analysis. It provides a way to express complex numbers in an exponential form, which is highly advantageous in calculations involving rotation and oscillation. The formula states that any complex number can be expressed as:$$e^{i\theta }=\cos (\theta )+i\sin (\theta )$$This means that a complex number can be viewed as a rotation in the complex plane. When a complex number is in the form $r{e}^{i\theta }$, it represents a circle of radius $r$ rotated by an angle $\theta$ from the positive real axis.
By using Euler's Formula, complex numbers can be easily converted between rectangular (algebraic) form $a+ib$ and polar (trigonometric) form $r(\cos (\theta )+i\sin (\theta ))$. This is particularly useful for simplifying the multiplication, division, and exponentiation of complex numbers, as these operations become straightforward in the exponential form. Euler's Formula thus bridges the gap between algebraic and geometric representations.

Polar and Rectangular Form Conversion

Converting between polar and rectangular forms is an essential skill in handling complex numbers. - In the rectangular form, a complex number is expressed as $z=a+ib$, where $a$ is the real part and $b$ is the imaginary part. This form is very intuitive and useful for addition and subtraction.- In polar form, the same number becomes $z=r{e}^{i\theta }$, where $r$ is the magnitude or absolute value, and $\theta$ is the argument or angle. Polar form is conducive for multiplication, division, and exponentiation.The conversion is done via these relationships:$$r=\sqrt{a^2+b^2}$$ and $$\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$$ for moving from rectangular to polar form. To convert back, use:$$a=r\cos (\theta )$$ and $$b=r\sin (\theta )$$.
This method allows seamless transition between the forms, making computations easier depending on the mathematical operations being performed.

Trigonometric Functions

Trigonometric Functions play a critical role in complex number expressions and conversions. They form the backbone of representing complex numbers in polar form through Euler’s Formula.- **Cosine (cos)** and **Sine (sin)** are used to map the real and imaginary parts of a complex number in polar form. For an angle $\theta$, the point on the unit circle is represented by $\cos (\theta )+i\sin (\theta )$.- These functions help in determining the angle $\theta$ in radians, a necessary step for the polar representation. Knowing the specific values of sine and cosine for standard angles is crucial, as this aids in simplifying expressions.- When working through problems such as converting $4e^{i\pi /6}$ into rectangular form, knowledge of trigonometric values allows you to extract and manipulate the components $\cos (\pi /6)$ and $\sin (\pi /6)$.Understanding these functions is vital, not only for conversion but also for other applications in calculus, physics, and engineering where complex numbers are utilized.