Problem 114 - 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. $$\left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4}$$

Step-by-Step Solution

Answer

The complex number \(\left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4}\) in standard form is \(-40.5 + 70.5\sqrt{3}i\).

Step 1: Express the Complex Number in Polar Form

The given complex number is in the form \(r(\cos \theta + i \sin \theta)\), where \(r = 3\) and \(\theta = 150^{\circ}\) or in terms of radians \(\theta = \frac{5\pi}{6}\). We are asked to find the forth power of this complex number.

Step 2: Apply DeMoivre's Theorem

According to DeMoivre's Theorem, to raise a complex number to a power \(n\), the modulus \(r\) is raised to the power \(n\) and the angle \(\theta\) is multiplied by \(n\). Here, \(n = 4\). Therefore, the new modulus \(r' = r^n = 3^4 = 81\), and the new angle \(\theta' = n\theta = 4 \times \frac{5\pi}{6} = \frac{20\pi}{6}\). This simplifies to \(\theta' = \frac{10\pi}{3}\).

Step 3: Express the Result in Standard Form

In order to express the result in standard form \(a + bi\), we need to find the Cartesian coordinates of the point representing the complex number on the complex plane. Given \(r' = 81\) and \(\theta' = \frac{10\pi}{3}\), we have \(a = r' \cos \theta' = 81 \cos(\frac{10\pi}{3}\) and \(b = r' \sin \theta' = 81 \sin(\frac{10\pi}{3}\). The cosine and sine of \(\frac{10\pi}{3}\) are \(-\frac{1}{2}\) and \(\frac{\sqrt{3}}{2}\) respectively. Therefore, \(a = 81 \times -\frac{1}{2} = -40.5\) and \(b = 81 \times \frac{\sqrt{3}}{2} = 70.5\sqrt{3}\).

Key Concepts

Complex Numbers

Complex numbers are fundamental components of advanced mathematics, particularly when dealing with equations that cannot be solved using real numbers alone. A complex number consists of two parts: a real part and an imaginary part. It is typically expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the property that i^2 = -1.

For example, in the complex number 4 + 3i, 4 is the real part and 3 is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but their unique properties enable the solution of equations that have no real solutions, such as x^2 + 1 = 0.

Polar Form of Complex Numbers

The polar form of a complex number is another way to represent complex numbers, emphasizing their geometric interpretation. Instead of using the real and imaginary parts, the polar form uses a magnitude (also called modulus) and an angle. The magnitude r of the complex number corresponds to its distance from the origin on the complex plane, while the angle θ (theta) represents the counterclockwise rotation needed from the positive real axis to the line that connects the origin to the number.

The general expression for a complex number in polar form is r(cos θ + i sin θ), or using Euler's formula, r*e^(iθ). Polar form is particularly useful when performing multiplication or finding powers of complex numbers, as it simplifies the process into multiplying magnitudes and adding angles.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is an alternative expression to the polar form, which utilizes trigonometric functions to describe the position of a complex number. Given a complex number's polar form r(cos θ + i sin θ), the trigonometric form simply spells this out using trigonometric functions. In computations, it often simplifies processes including the multiplication, division, and finding roots of complex numbers.

Understanding the trigonometric form can greatly assist in visualizing complex numbers on the complex plane, where the cosine function relates to the x-coordinate (real part) and the sine function relates to the y-coordinate (imaginary part). This visual representation is handy when we want to find powers or roots as it relates to rotating and scaling vectors in the plane.

Powers of Complex Numbers

Computing the powers of complex numbers can seem daunting at first, but it becomes more approachable with DeMoivre's Theorem. This theorem states that to raise a complex number in polar or trigonometric form to a power n, you simply raise the modulus r to the power of n and multiply the angle θ by n. The resulting complex number is r^n times (cos(nθ) + i sin(nθ)).

For instance, to find the fourth power of 3(cos 150° + i sin 150°), you raise 3 to the power of 4 to get the new modulus 81, and multiply 150° by 4 to get the new angle 600°, which can be reduced to its equivalent angle within a full circle. This makes calculations involving complex numbers in polar or trigonometric form much more manageable and is especially powerful when dealing with higher powers or roots of complex numbers.