Problem 13 - Contemporary Precalculus

Question

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$\left(\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{14}$$

Step-by-Step Solution

Answer

Based on the given problem and solution, the short answer question can be: Question: Calculate the product of the complex number \(\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\) raised to the power of 14. Answer: \(0+i\)

Step 1: Convert Complex Number to Polar Form

In order to convert the given complex number to polar form, we can use the formula: $$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$$ where \(r = \sqrt{a^2 + b^2}\) and \(\theta = \arctan(\frac{a}{b})\), with \(a\) and \(b\) being the real and imaginary parts respectively. First, we calculate \(r\) and \(\theta\) for our given complex number \(\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\): $$r = \sqrt{\left(\frac{-1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = 1$$ $$\theta = \arctan\left(\frac{1}{-1}\right) = \arctan(-1) = \frac{3\pi}{4}$$ So our complex number in polar form is \(1(\cos(3\pi/4) + i\sin(3\pi/4))\).

Step 2: Apply DeMoivre's Theorem

Now we apply DeMoivre's Theorem to raise our complex number to the power of 14: $$\left[1(\cos(3\pi/4) + i\sin(3\pi/4))\right]^{14}$$ DeMoivre's Theorem states that for any complex number in polar form \(z=re^{i\theta}\) and any integer n: $$(z)^n = r^n[\cos(n\theta)+i\sin(n\theta)]$$ Applying this theorem, we have: $$\left[1(\cos(3\pi/4) + i\sin(3\pi/4))\right]^{14} = 1^{14}\left[\cos\left(\frac{42\pi}{4}\right) + i\sin\left(\frac{42\pi}{4}\right)\right]$$

Step 3: Convert Back to Cartesian Form

Now we convert the result back to Cartesian form. $$1\left[\cos\left(\frac{42\pi}{4}\right) + i\sin\left(\frac{42\pi}{4}\right)\right] = 1(\cos(10\pi + \pi/2) + i\sin(10\pi + \pi/2))$$ Using the trigonometric identities: $$\cos(10\pi + \pi/2) = 0$$ $$\sin(10\pi + \pi/2) = 1$$ So the result is: $$1(0 + i\cdot 1) = 0+i$$ Hence, the final answer is \(0+i\).

Key Concepts

Polar Form of Complex Numbers

To understand the process demonstrated in the exercise, it is essential to first grasp what the polar form of a complex number is. A complex number, typically written in Cartesian form as a + bi, where a is the real part, and b is the imaginary part, can also be expressed in polar form.

This is particularly useful when dealing with complex number multiplication, division, and exponentiation, as it allows us to use the magnitude and angle of the number rather than the individual components. The polar form of a complex number uses the formula \(z = re^{i\theta} = r(\cos\theta + i\sin\theta)\), where \(r\) is the magnitude (modulus) of the complex number, calculated as \(\sqrt{a^2 + b^2}\), and \(\theta\) is the argument (angle), found using the arctan function of b/a.

The exercise demonstrates converting the given complex number into the polar form which simplifies the process of raising it to a power, as shown in DeMoivre's Theorem.

Trigonometric Identities

The exercise also makes use of trigonometric identities to simplify the calculations. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. Some of the most commonly used identities include \(\sin^2(\theta) + \cos^2(\theta) = 1\) and the angle sum and difference formulas which express \(\sin(\theta \pm \phi)\) and \(\cos(\theta \pm \phi)\) in terms of \(\sin(\theta)\), \(\cos(\theta)\), \(\sin(\phi)\), and \(\cos(\phi)\).

In the provided solution, after raising the complex number to the 14th power using DeMoivre's Theorem, the result is simplified using identities such as \(\cos(2k\pi + \frac{\theta}{2}) = 0\) and \(\sin(2k\pi + \frac{\theta}{2}) = 1\) for any integer k. These identities greatly facilitate the process of reverting the complex number back to its rectangular or Cartesian form.

Complex Numbers Exponentiation

When exponentiating complex numbers, as in the given exercise, DeMoivre's Theorem is an indispensable tool. This theorem states that for any positive integer n, the nth power of a complex number \(z\) in polar form \(re^{i\theta}\) or \(r(\cos\theta + i\sin\theta)\) is \( r^n(\cos(n\theta) + i\sin(n\theta))\).

In simpler terms, to raise a complex number to a power, you raise the magnitude r to that power and multiply the angle \theta by that power. This principle turns a potentially complex multiplication into a more straightforward operation involving the radius and angle of the complex number. As demonstrated in the exercise, after applying DeMoivre's Theorem, the complex number is easily raised to the 14th power, and by applying trigonometric identities, we can quickly find the final solution in Cartesian form.