Problem 74 - 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. \([3(\cos\ 60^{\circ} + i\ \sin\ 60^{\circ})]^4\)

Step-by-Step Solution

Answer

The indicated power of the complex number in standard form is -40.5 - 70.5i.

Step 1: Apply DeMoivre's Theorem

To start the application of DeMoivre's Theorem to the complex number, plug \( r = 3 \), \( θ = 60^{\circ} \), and \( n = 4 \) into the theorem as follows: \( (r (\cos θ + i \sin θ))^n = r^n (\cos (nθ) + i\sin (nθ)) \). This gives us \( [3 (\cos 60^{\circ} + i \sin 60^{\circ})]^4 = 3^4 (\cos 4*60^{\circ} + i \sin 4*60^{\circ}) \).

Step 2: Compute the power and the angle

Next, calculate the value of \( 3^4 \) and \( 4*60^{\circ} \). That gives us 81 and 240 degrees respectively. This gives us \( 81 (\cos 240^{\circ} + i \sin 240^{\circ}) \)

Step 3: Convert to standard form

Now that we have the magnitude and the angle, we can convert the complex number to standard form. This involves replacing \( \cos 240^{\circ} \) with -1/2 and \( \sin 240^{\circ} \) with \( -\sqrt{3}/2 \). Thus, we get \( 81 [(-1/2) + i (-\sqrt{3}/2)] = -40.5 - 70.5i \), which is in standard form.

Key Concepts

Complex Number

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). The term \(a\) is called the real part, while \(bi\) is the imaginary part. Complex numbers extend the idea of one-dimensional real numbers into the two-dimensional complex plane, by using the horizontal axis for the real part and the vertical axis for the imaginary part. This allows for many operations, such as addition, subtraction, and multiplication.

• Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
• Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
• Multiplication: \((a + bi)(c + di) = ac + bci + adi + bdi^2 = (ac - bd) + (bc + ad)i\)

This foundational understanding is vital when utilizing DeMoivre’s Theorem for calculating powers of complex numbers expressed in polar form.

Standard Form

The standard form of a complex number is written as \(a + bi\). It is a straightforward way to express complex numbers and is widely used in calculations and representations. In standard form:

• \(a\) is the real part.
• \(bi\) is the imaginary part, where \(i\) is the imaginary unit \(\sqrt{-1}\).

Any complex number can be transformed into standard form from different representations such as polar form. For instance, given polar coordinates or after using DeMoivre’s Theorem, we derive the standard form by calculating the cosine and sine components of the argument and multiplying them by the modulus. This transformation is crucial in exercises that involve writing the result in standard form after performing operations using DeMoivre's Theorem.

Polar Coordinates

Polar coordinates are another way to represent complex numbers, where a complex number \(z\) is represented as \(r(\cos \theta + i \sin \theta)\). Here, \(r\) is the magnitude or modulus of the complex number and \(\theta\) is the argument or angle.
This form is very useful for multiplying and dividing complex numbers, as well as for finding powers and roots, because:

• Multiplication and division become simpler because of the properties of exponents.
• Angles add when multiplying, and subtract when dividing.

For example, to find the power of the complex number using DeMoivre's Theorem, we use the polar coordinates where the magnitude is raised to a power and the angle is multiplied by the power. This simplification is particularly useful in solving exercises like finding \([3(\cos 60^\circ + i \sin 60^\circ)]^4\).

Power of a Complex Number

Finding the power of a complex number can be simplified using DeMoivre's Theorem. DeMoivre’s Theorem states that for a complex number \(z = r(\cos \theta + i \sin \theta)\), its \(n\)-th power can be given by \((r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i \sin(n\theta))\). This theorem effectively transforms a complex number raised to a power into a manageable form by:

• Raising the magnitude \(r\) to the power \(n\).
• Multiplying the angle \(\theta\) by \(n\).

In practice, after computing the new modulus and angle, it is important to convert this back to the standard form. For example, in the exercise provided, \([3(\cos 60^\circ + i \sin 60^\circ)]^4\), we computed the power as \(81(\cos 240^\circ + i \sin 240^\circ)\) and then converted it to standard form: \(-40.5 - 70.5i\). This conversion is important to interpret the result correctly in the context of complex numbers.