Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right]^{4}$$

Step-by-Step Solution

Answer

The power in rectangular form is \(-81\).

Step 1: Convert Polar Form to Rectangular Form

The given complex number is in polar form: \( 3 \left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right) \). The polar form \( z = r(\cos \theta + i \sin \theta) \) can be converted to rectangular form using \( x = r \cos \theta \) and \( y = r \sin \theta \).For \( z = 3 \left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right) \):- Calculate \( x = 3 \cos \frac{3\pi}{4} = 3 \left( -\frac{\sqrt{2}}{2} \right) = -\frac{3\sqrt{2}}{2} \).- Calculate \( y = 3 \sin \frac{3\pi}{4} = 3 \left( \frac{\sqrt{2}}{2} \right) = \frac{3\sqrt{2}}{2} \).Thus, the rectangular form is: \( -\frac{3\sqrt{2}}{2} + i \frac{3\sqrt{2}}{2} \).

Step 2: Apply De Moivre's Theorem

To find the power of a complex number in polar form, we can use De Moivre's Theorem: \( [r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta)) \).Here, \( r = 3 \), \( \theta = \frac{3\pi}{4} \), and \( n = 4 \).- Calculate \( r^4 = 3^4 = 81 \).- Calculate \( n\theta = 4 \times \frac{3\pi}{4} = 3\pi \).So we have: \( 81 \left( \cos 3\pi + i \sin 3\pi \right) \).

Step 3: Simplify Using Trigonometric Identities

Next, we determine \( \cos 3\pi \) and \( \sin 3\pi \) using trigonometric identities:- \( \cos 3\pi = \cos(2\pi + \pi) = -1 \), because \( \cos(\pi) = -1 \).- \( \sin 3\pi = \sin(2\pi + \pi) = 0 \), because \( \sin(\pi) = 0 \).Substituting these values into the expression, we get: \( 81( -1 + i \cdot 0 ) = -81 \).

Step 4: Write the Answer in Rectangular Form

The answer obtained from simplifying is \( -81 + 0i \), which is simply \( -81 \). This is already in the rectangular form: \((x + yi)\). Hence, the rectangular form of the given power is \( -81 \).

Key Concepts

Polar Form

In complex numbers, the polar form is a way of expressing a number using the angle and distance from the origin on the complex plane. It's like having a GPS coordinate for your complex number. Similar to how you would say that you're 10 miles northeast of a location, you express your complex number as a distance (magnitude) and an angle (direction). This form is typically written as: \[ z = r(\cos \theta + i \sin \theta) \] where \( r \) is the magnitude of the complex number, and \( \theta \) is the angle it makes with the positive real axis. The magnitude, \( r \), is found using the formula \( r = \sqrt{x^2 + y^2} \), and the angle, \( \theta \), is calculated using \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \).

• Polar form is particularly useful for multiplying and dividing complex numbers, as the magnitudes are multiplied or divided and the angles are added or subtracted.
• It simplifies the process of raising complex numbers to powers or finding roots.

Rectangular Form

Rectangular form is the standard way of writing a complex number using real and imaginary components. It's the form where you might jot down a number like \( a + bi \) in your math notebook. Here, \( a \) is the real part and \( b \) is the imaginary part. This form is fantastic for adding and subtracting complex numbers, as it mirrors the way you handle regular algebraic expressions.

• The rectangular form looks like \( z = x + yi \), where \( x \) and \( y \) are real numbers, representing the real and imaginary components, respectively.
• Converting from polar to rectangular involves using the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \).

In the provided solution, this was shown as converting a polar expression like \( 3(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) \) to its rectangular counterpart through simple trigonometric calculations.

De Moivre's Theorem

This handy theorem helps us find powers and roots of complex numbers in polar form. It's like a magic key for simplifying complicated calculations. De Moivre's Theorem states that for a complex number in the form \( r(\cos \theta + i \sin \theta) \), its \( n \)-th power is computed as: \[ [r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta)) \] This is remarkably efficient because it allows you to multiply the angle and raise the magnitude to the power, rather than handling the complex multiplication of a rectangular form.

Practical Uses

• Used extensively in signal processing, physics, and anywhere complex numbers need raising to powers.
• Simplifies calculations by breaking them down into manageable parts.

In the solution provided, this theorem allowed conversion from \( 3(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})^4 \) directly to \( 81(\cos 3\pi + i \sin 3\pi) \).

Trigonometric Identities

These are the mathematical shortcuts that make working with angles much easier. They are formulas that relate the angles of a triangle to the lengths of its sides and can convert trigonometric forms into simpler, more manageable expressions. Common identities include: \[ \cos(\theta + 2\pi k) = \cos \theta \] \[ \sin(\theta + 2\pi k) = \sin \theta \] These identities allow you to handle angles greater than \( 2\pi \) by wrapping them back into their base circle.

Useful for recalculating trigonometric values without re-doing the entire unit circle navigation.

Facilitates the simplification of complex expressions involving periodic functions.

In our exercise, trigonometric identities simplify the evaluations of \( \cos 3\pi \) and \( \sin 3\pi \), directing the original polar expression \( 81(\cos 3\pi + i \sin 3\pi) \) simply to the real number \( -81 \).