Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)^{10}$$

Step-by-Step Solution

Answer

The power of the complex number \( \left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)^{10} \) in standard form is \( i \).

Step 1: Express the given complex number in polar form and state the power

The complex number is given by \( \left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)^{10} \) . In the polar form, this complex number is represented as \( r(\cos \theta + i \sin \theta) \) where \( r \) is the modulus and \( \theta \) is the argument. Since cosine and sine already given, we can say that \( r = 1 \) and \( \theta = \frac{5 \pi}{4} \). The power we need to find is \( n = 10 \).

Step 2: Apply DeMoivre's theorem

According to De Moivre's theorem, \( (r(\cos \theta + i \sin \theta))^n = r^n(\cos(n \theta) + i \sin(n \theta)) \). As \( r = 1 \) and \( n \theta = 10 \cdot \frac{5 \pi}{4} = \frac{25 \pi}{2} \), our result becomes \( (1)^{10} (\cos(\frac{25 \pi}{2}) + i \sin(\frac{25 \pi}{2})) \). Which simplify to \( \cos(\frac{25 \pi}{2}) + i \sin(\frac{25 \pi}{2}) \). To proceed further, we need to use the periodicity of sine and cosine function. As they have a period of \( 2 \pi \), \( \cos( \frac{25 \pi}{2}) \) is equal to \( \cos(\frac{\pi}{2}) \) and \( \sin(\frac{25 \pi}{2}) \) is equal to \( \sin(\frac{\pi}{2}) \). So it simplifies to \( \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \).

Step 3: Express the Result in Standard Form

In standard form, \( \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \) is equal to \( 0 + i \cdot 1 \), So, the power of the complex number in standard form is \( i \).

Key Concepts

Complex Numbers

Complex numbers are an essential part of advanced mathematics, combining both real numbers and imaginary numbers to form a more comprehensive number system. A complex number is usually represented as \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as the square root of \( -1 \), offering a way to perform algebraic operations with negative square roots.

These numbers can be particularly useful in fields such as engineering and physics, especially when dealing with waves or oscillations. In the realm of mathematics, complex numbers allow for the finding of roots in equations that cannot be solved using just real numbers.

Understanding the properties of complex numbers is crucial because it builds the foundation for expressing them in other forms like the polar form — a key step in utilizing the DeMoivre's theorem.

Polar Form

Polar form is a way to express complex numbers using a different coordinate system which is very insightful when dealing with rotations and periodic functions. Instead of using the usual Cartesian coordinates, polar form uses a radius \( r \) and an angle \( \theta \) to describe a complex number.

In polar form, a complex number is represented as \( r(\cos \theta + i \sin \theta) \). The modulus \( r \) indicates the distance from the origin to the point in the complex plane, while \( \theta \) represents the angle from the positive x-axis to the line connecting the origin to the point.

Converting a complex number from standard form \( a + bi \) to polar form involves calculating the modulus \( r = \sqrt{a^2 + b^2} \) and the argument \( \theta = \tan^{-1}(\frac{b}{a}) \). Once in polar form, complex operations such as multiplication, division, and exponents become far more straightforward because of the neat association with angle and radius.

Trigonometric Form

Trigonometric form is essentially a version of the polar form for complex numbers but is more explicit about its use of trigonometric functions. In trigonometric form, a complex number is expressed as \( r \cdot (\cos\theta + i\sin\theta) \), highlighting how the trigonometric identities tie into complex numbers.

One of the primary benefits of using trigonometric form is the ease of raising a complex number to a power, especially leveraging DeMoivre's Theorem. This theorem simplifies the process to \( (r (\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \). This is invaluable for calculations involving powers or roots of complex numbers.

This form is instrumental in simplifying problems in calculus and complex analysis by transforming them into rotational problems in the complex plane, thus bridging the gap between algebraic and geometric interpretations.