Problem 64 - Algebra and Trigonometry

Question

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$ (\sqrt{2}-i)^{4} $$

Step-by-Step Solution

Answer

The fourth power of \(\sqrt{2}-i\) in rectangular form is \(9*(cos(-4*tan^{-1}(1/\sqrt{2})) - i*sin(4*tan^{-1}(1/\sqrt{2})))\)

Step 1: Converting from Rectangular to Polar Form

First step is to convert the given complex number \(\sqrt{2}-i\) from rectangular form to polar form. To find radius \(r\), we can use the formula: \(r = \sqrt{a^2 + b^2}\) where \(a\) and \(b\) are the real and imaginary parts of the complex number respectively. So here, \(a=\sqrt{2}\) and \(b=-1\). Hence, \( r=\sqrt{(\sqrt{2})^{2}+(-1)^{2}} = \sqrt{2+1}= \sqrt{3}\). Now, we need to find the argument \(θ\). Since \(a\) is positive and \(b\) is negative, we will be in the fourth quadrant, hence \(θ\) can be calculated as: \(θ = 2\pi - tan^{-1}\left|\frac{b}{a}\right|\), which is \(2\pi - tan^{-1}(1/\sqrt{2})\)

Step 2: Applying DeMoivre's Theorem

DeMoivre's theorem states that \((re^{iθ})^{n} = r^{n}e^{inθ}\) . So let's apply this to our complex number. Here, r=\(\sqrt{3}\), θ= \(2\pi - tan^{-1}(1/\sqrt{2})\), and n=4. Hence \((re^{iθ})^{4} = ( \sqrt{3} e^{i(2\pi - tan^{-1}(1/\sqrt{2}))})^{4}= 3^{2}e^{4i(2\pi - tan^{-1}(1/\sqrt{2}))}\). Using the fact that \(e^{2\pi i}=1\), we simplify the expression further to get: \(9e^{4i(- tan^{-1}(1/\sqrt{2}))}\)

Step 3: Convert back to Rectangular form

The final step is to convert our answer back into rectangular form. This uses the Euler's formula which states that \(e^{ix}= cosx + i*sinx\). Therefore, our expression becomes: 9[(cos(4*(- tan^{-1}(1/\sqrt{2})))) + i * sin(4*(- tan^{-1}(1/\sqrt{2})))] which simplifies to: 9(cos(-4*tan^{-1}(1/\sqrt{2}))) - 9i*sin(4*tan^{-1}(1/\sqrt{2}))

Key Concepts

Polar Form

The polar form of a complex number is a way to express a complex number in terms of its magnitude and angle rather than real and imaginary components. Every complex number can be written as:

• The magnitude, also known as radius, is the distance from the origin to the point, calculated as \( r = \sqrt{a^2 + b^2} \) where \(a\) and \(b\) represent the real and imaginary parts, respectively.
• The angle, or argument, \( \theta \), is the direction of the line from the origin. It is often found using inverse trigonometric functions. For example, in the fourth quadrant, \( \theta = 2\pi - \tan^{-1}\left|\frac{b}{a}\right|\).

The polar form is particularly useful in simplifying the multiplication and division of complex numbers, as well as in finding powers, due to the use of Euler's identity, \( e^{i\theta} = \cos \theta + i\sin \theta \). By writing a complex number like \( \sqrt{2}-i \) in polar form, calculations involving powers become straightforward to perform with DeMoivre's Theorem.

Rectangular Form

Rectangular form is the most familiar way to express complex numbers, denoted as \( a + bi \), where \(a\) is the real part and \(bi\) is the imaginary part. This form is useful for addition and subtraction of complex numbers because you can directly add or subtract the corresponding real parts and the imaginary parts.

• Changing from polar to rectangular form involves finding \(a\) and \(b\) using \( r \cos\theta \) and \( r \sin\theta \) respectively.
• For instance, when we have the polar form result from a calculation, such as \( re^{i\theta} \), we convert it back to \( a + bi \) form by applying Euler's formula: \( re^{i\theta} = r(\cos \theta + i\sin \theta) \).

Rectangular form helps visualize a complex number on the complex plane, making it clear which part represents the real number axis and which represents the imaginary.

Complex Numbers

Complex numbers expand the number system to include solutions to equations like \( x^2 + 1 = 0 \). They have a real part and an imaginary part, denoted as \( a + bi \) where \(i\) is the imaginary unit with the property that \(i^2 = -1\).

• Complex numbers are essential in various fields, like engineering and physics, as they can capture phenomena that real numbers cannot.
• In addition to addition and subtraction, complex numbers can be treated algebraically: multiplying involves distributing terms and using \(i^2 = -1\), while division involves multiplying by the conjugate to rationalize the denominator.

Complex numbers are plotted on a complex plane similar to the Cartesian coordinate system, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Understanding both polar and rectangular forms is crucial to leverage the full power of complex numbers.