Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{4}+16=0$$

Step-by-Step Solution

Answer

The solutions are \[ x_1 = 2 \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \], \[ x_2 = 2 \left( \cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right) \], \[ x_3 = 2 \left( \cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right) \], and \[ x_4 = 2 \left( \cos \frac{9\pi}{4} + i \sin \frac{9\pi}{4} \right) \].

Step 1: Rewrite the Equation

The given equation is \(x^4 + 16 = 0\). We can rewrite it as \(x^4 = -16\). This rearranges the equation into a form that allows us to find the roots.

Step 2: Express as Powers of i

Note that \(-16\) can be expressed as \(-16 = 16 \cdot (-1)\). Note that \(-1 = i^2\), therefore we can write \(-16 = 16i^2\). Our equation thus is \(x^4 = 16i^2\).

Step 3: Take the Fourth Root

To solve \(x^4 = 16i^2\), we take the fourth root of both sides: \(x = (16i^2)^{1/4}\). This simplifies to \(x = (16^{1/4})(i^2)^{1/4}\).

Step 4: Simplify the Fourth Root

The fourth root of 16 is 2, and using De Moivre's theorem, the fourth root of \(i^2=-1\) gives the complex numbers evenly distributed on the unit circle. Each root is separated by \(\frac{\pi}{2}\) (or 90 degrees) starting from \(-\frac{\pi}{2}\).

Step 5: Identify the Roots

Since we need the fourth root of \(i^2 = -1\), the roots are \(\pm(\sqrt{2}) \left( \cos\left(\frac{3\pi}{4} + n\pi\right) + i \sin\left(\frac{3\pi}{4} + n\pi\right) \right)\) for \(n=0,1,2,3\). These roots represent the solutions midnight beautifully distributed in the complex plane.

Step 6: Write the Roots in Trigonometric Form

Convert the solutions found into trigonometric form: - \(x_1 = 2 \left(\cos\left(\frac{3\pi}{4}\right) + i \sin\left(\frac{3\pi}{4}\right)\right)\) - \(x_2 = 2 \left(\cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right)\right)\) - \(x_3 = 2 \left(\cos\left(\frac{7\pi}{4}\right) + i \sin\left(\frac{7\pi}{4}\right)\right)\) - \(x_4 = 2 \left(\cos\left(\frac{9\pi}{4}\right) + i \sin\left(\frac{9\pi}{4}\right)\right)\)

Key Concepts

Trigonometric Form

When dealing with complex numbers, especially in solving equations, the trigonometric form can be very useful. This form expresses complex numbers using angles and magnitudes. A complex number \( z \) can be written as \( z = r(\cos\theta + i\sin\theta) \), where:

• \(r\) is the modulus, or the distance from the origin to the point on the complex plane.
• \(\theta\) is the argument, or the angle the line makes with the positive real axis.

This representation is particularly powerful because it reveals the geometric nature of complex numbers.
It shows how they "rotate" around the origin and how far they extend from it.
It's also foundational for multiplying and dividing complex numbers, as these operations simply add or subtract their respective angles when numbers are in trigonometric form.

Roots of Complex Numbers

Finding roots of complex numbers involves finding all numbers that, when raised to a specific power, yield the original complex number. For example, finding the fourth roots of a number means identifying four numbers that when raised to the fourth power result in the original number.
These roots are particularly interesting because they are evenly spaced around the complex plane.

• This means each root is separated by equal angles.
• They form a geometric progression in their placement.

If the number is expressed in trigonometric form \( r(\cos(\theta) + i\sin(\theta)) \), the \( n \)-th root can be found as:\[ z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \] where \(k = 0, 1, ..., n-1\). This formula makes finding roots manageable and visually interpretable.

De Moivre's Theorem

De Moivre's Theorem provides a bridge between complex numbers and trigonometry.
It offers a way to compute powers and roots of complex numbers in trigonometric form easily.
The theorem states:

• For any real number \( \theta \) and integer \( n \), \[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \]

This theorem is particularly useful because it allows us to compute powers and roots of complex numbers by simply multiplying or dividing the angles:

• To raise a complex number to the \( n \)-th power, you multiply the argument by \( n \).
• To find the \( n \)-th root, you divide the argument by \( n \), distributing the solutions evenly around the circle.

This characteristic is harnessed in solving equations like \( x^4 + 16 = 0 \), where the roots are expressed using De Moivre's Theorem to be evenly spaced angles around the complex plane.