Problem 80 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (1-i)^{-8} $$

Step-by-Step Solution

Answer

The power is \(\frac{1}{16}\).

Step 1: Convert to Polar Form

First, express the complex number \(1-i\) in polar form. A complex number \(a+bi\) can be expressed in polar form as \(r(\cos \theta + i\sin \theta)\), where \(r = \sqrt{a^2 + b^2}\) and \(\theta = \tan^{-1}(\frac{b}{a})\). For \(1-i\), \(a=1\) and \(b=-1\). Calculate \(r\): \[r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\]Calculate \(\theta\): \[\theta = \tan^{-1}\left(-1\right) = -\frac{\pi}{4}\]Thus, \(1-i = \sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form \(r (\cos \theta + i \sin \theta)\), \[(r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))\]Set \(n = -8\), \(r = \sqrt{2}\), and \(\theta = -\frac{\pi}{4}\).Then, \[(\sqrt{2})^{-8}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))^{-8} = (\sqrt{2})^{-8} \left(\cos(-8 \times \frac{\pi}{4}) + i \sin(-8 \times \frac{\pi}{4})\right)\]

Step 3: Simplify the Expression

Calculate \((\sqrt{2})^{-8}\):\[(\sqrt{2})^{-8} = (2)^{-4} = \frac{1}{16}\]Calculate \(8 \times \frac{\pi}{4}\): \[8 \times \frac{\pi}{4} = 2\pi\]Therefore:\[\cos(-2\pi) = \cos(0) = 1\]\[\sin(-2\pi) = \sin(0) = 0\]Plug these values back into the equation:\[(\sqrt{2})^{-8}(\cos(-2\pi) + i\sin(-2\pi)) = \frac{1}{16}(1 + 0i) = \frac{1}{16}\]

Step 4: Conclusion

Thus, the power of \((1-i)^{-8} = \frac{1}{16}\). This solution uses conversion to polar coordinates and De Moivre's Theorem to simplify the calculation.

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical objects that extend our familiar concept of numbers. They are composed of two parts: a real part and an imaginary part. The general form is expressed as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The "\(i\)" represents the imaginary unit, which is defined by \(i^2 = -1\).
Complex numbers allow us to perform operations such as addition, subtraction, multiplication, and division, similar to real numbers. Moreover, they enable solving equations that have no real solutions, like \(x^2 + 1 = 0\).

• Complex numbers are visualized on the complex plane with the x-axis as the real part and the y-axis as the imaginary part.
• The length from the origin to the point \((a, b)\) on the plane is the magnitude of the complex number, calculated using \(r = \sqrt{a^2 + b^2}\).

Polar Coordinates

Polar coordinates provide a unique way to represent complex numbers, a concept especially useful that simplifies multiplication and exponentiation of complex numbers. Instead of using rectangular coordinates (real and imaginary parts), polar coordinates use a radius and an angle.

• The radius \(r\) is the distance of the point from the origin, computed as \(r = \sqrt{a^2 + b^2}\).
• The angle \(\theta\), known as the argument, is the direction of the radius, calculated as \(\theta = \tan^{-1}(\frac{b}{a})\).

In polar form, a complex number is expressed as \(r(\cos \theta + i\sin \theta)\). This representation is incredibly beneficial because it turns complex multiplication into a simpler operation of multiplying magnitudes and adding angles. Polar coordinates make De Moivre's Theorem possible, which is crucial for calculating powers and roots of complex numbers.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles of triangles. It plays a significant role in expressing complex numbers in polar form. The polar form of complex numbers relies on the trigonometric functions cosine and sine. By solving the angles using trigonometric concepts, we manage to simplify complex number computations.

• Cosine \(\cos \theta\) and sine \(\sin \theta\) help determine the direction of the complex number on the plane.
• When applied to complex numbers, sinusoids reveal cycles within the functions, such as those observed in the solution using \(\cos(2\pi) = 1\) and \(\sin(2\pi) = 0\).

Through trigonometric identities, solving powers or roots of complex numbers via De Moivre's Theorem becomes systematic, letting us predict periodic results when angles extend beyond 360 degrees or \(2\pi\) radians.

Exponents and Powers

Exponents and powers are foundational concepts in mathematics that describe repeated multiplication. In the context of complex numbers, they allow us to simplify intricate expressions using rules like De Moivre's Theorem.

• Raising a number to a power means multiplying it by itself several times, a straightforward operation with real numbers but more complex for complex numbers.
• Exponents are intimately related to logarithms and exponential functions, powerful tools across various fields of study.

De Moivre's Theorem bridges exponents and complex numbers by simplifying the process of raising polar form expressions of complex numbers to a power \(n\): \[(r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))\].
This method shrinks a potentially cumbersome task into more manageable steps, demonstrating the beauty and efficiency of mathematical structures.