Problem 81 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([3(\cos\ 15^{\circ}\ +\ i\ \sin\ 15^{\circ}]^4\)

Step-by-Step Solution

Answer

The result in standard form is \(40.5 + 35.35i\)

Step 1: Recognize and Apply DeMoivre's Theorem

We can use DeMoivre's theorem directly on the expression. Given \((\cos\ \theta + \ i\sin\ \theta)^n\), by DeMoivre's theorem, this transforms into: \(\cos\ n\theta +\ i\ \sin\ n\theta\). \n So, the given expression \([3(\cos\ 15^{\circ}\ +\ i\ \sin\ 15^{\circ}]^4\) becomes \(3^4(\cos\ 4\times15^{\circ}\ +\ i\ \sin\ 4\times15^{\circ}) = 81(\cos\ 60^{\circ}\ +\ i\ \sin\ 60^{\circ})\).

Step 2: Convert Polar to Rectangular Form

Next, let's move convert the result from polar form to rectangular form. The conversion formula is: \(r(\cos\ \theta +\ i\sin\ \theta) = r \cos\ \theta +\ r i\sin\ \theta\). Applying this formula gives: \(81[ \cos\ 60^{\circ}\ +\ i\ \sin\ 60^{\circ}] = 81 \cos\ 60^{\circ}\ +\ 81i\ \sin\ 60^{\circ}\)

Step 3: Compute Final Result

Finally, compute the actual values. For \(\cos\ 60^{\circ}\) the value is 1/2 and for \(\sin\ 60^{\circ}\) the value is \(\sqrt{3}/2\). Thus the final result is \( 81(\frac{1}{2}) + 81i (\frac{\sqrt{3}}{2}) = 40.5 + 35.35i \)

Key Concepts

Complex Numbers

Complex numbers are numbers that include both a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) representing the imaginary unit. The imaginary unit \(i\) is defined by \(i^2 = -1\).
Complex numbers are essential in advanced mathematics because they allow for solutions to equations that don't have real solutions. This happens because multiplying the imaginary unit \(i\) by itself results in -1, thus permitting the creation of "imaginary" parts in mathematical calculations.
When dealing with complex numbers, it is possible to perform standard arithmetic operations such as addition, subtraction, multiplication, and division.

• **Addition and Subtraction:** Simply add or subtract the real parts and the imaginary parts separately.
• **Multiplication:** Use the distributive property and remember that \(i^2 = -1\).
• **Division:** Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Understanding complex numbers is crucial for many fields, including engineering and physics, because they describe phenomena that cannot be represented with only real numbers.

Polar Form

The polar form of a complex number is an alternative to the rectangular form that can be especially useful for multiplication, division, and exponentiation of complex numbers.
In polar form, a complex number is represented as \(r(\cos\ \theta + i\sin\ \theta)\), where \(r\) is the magnitude (or modulus) of the number, and \(\theta\) is the angle (or argument) associated with the number.
The magnitude \(r\) is found using the formula \(r = \sqrt{a^2 + b^2}\), while the argument \(\theta\) is found using \(\theta = \tan^{-1}(b/a)\).
Polar form makes certain operations easier:

• **Multiplication and Division:** Simply multiply or divide the magnitudes and add or subtract the angles.
• **Exponentiation:** Utilize DeMoivre's Theorem, which states that \((r(\cos\ \theta + i\sin\ \theta))^n = r^n(\cos\ n\theta + i\sin\ n\theta)\).

By using polar form, complex numbers can be easily manipulated in terms of angles and magnitudes, which simplifies the process of raising them to powers, as shown in our original exercise with DeMoivre's Theorem.

Rectangular Form

Rectangular form is the standard way of expressing complex numbers as \(a + bi\), where \(a\) represents the real component and \(b\) signifies the imaginary component.
Most often used for addition and subtraction, rectangular form directly displays the real and imaginary parts of a complex number.
To express a number in rectangular form from polar form, use the conversion equations:

• \(x = r\cos\theta\) for the real part
• \(y = r\sin\theta\) for the imaginary part

This transformation allows complex numbers expressed in polar form to be rewritten back into their real and imaginary components, as demonstrated when converting \(81(\cos\ 60^\circ + i\sin\ 60^\circ)\) to \(40.5 + 35.35i\).
Bear in mind, while rectangular form is straightforward for basic operations, it can be cumbersome for more complex operations like multiplication or division, where polar form offers advantages. Hence, understanding both forms and knowing when to use each is vital for working effectively with complex numbers.