Problem 57 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$-7\left(\cos 140^{\circ}+i \sin 140^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is \( 5.362 - 4.4996i \).

Step 1: Rewrite Using Euler's Formula

The expression is given in polar form: \[-7(\cos 140^{\circ} + i \sin 140^{\circ})\]. Recall that Euler's formula can represent this as \[re^{i\theta} = r(\cos \theta + i \sin \theta)\]. Here, \( r = -7 \) and \( \theta = 140^{\circ} \).

Step 2: Calculate the Real Component

The real component can be calculated using:\[ -7 \cos 140^{\circ} \].Using a calculator, \( \cos 140^{\circ} \approx -0.7660 \).Therefore, the real part is:\[ -7 \times (-0.7660) = 5.362 \].

Step 3: Calculate the Imaginary Component

The imaginary component can be calculated with:\[ -7 \sin 140^{\circ} \].Using a calculator, \( \sin 140^{\circ} \approx 0.6428 \).Thus, the imaginary part is:\[ -7 \times 0.6428 = -4.4996 \].

Step 4: Combine Real and Imaginary Components

Combining both components, the rectangular form of the complex number is:\[ 5.362 - 4.4996i \].Thus, the rectangular form is expressed as\( 5.362 - 4.4996i \).

Key Concepts

Understanding Complex Numbers

Complex numbers are fascinating elements in mathematics, embodying both a real and an imaginary part. They are written typically in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Here, \( i \) represents the imaginary unit, with the property \( i^2 = -1 \).

• Real Part: This is the component without the \( i \) or technically, multiplied by 1. It represents a number on the real number line.
• Imaginary Part: It involves the imaginary unit \( i \) and signifies a "dimension" that the real number line alone cannot capture.

Complex numbers are the comprehensive system that fills the gaps between real numbers, enabling solutions to equations too difficult to solve using only real numbers, like \( x^2 + 1 = 0 \). Here, \( x \) becomes \( i \).
In the exercise, we transformed a polar form to this rectangular expression form, highlighting the real and imaginary aspects: \( 5.362 - 4.4996i \).

Exploring Polar Coordinates

Polar coordinates offer a different way to understand complex numbers using a magnitude and angle rather than the traditional coordinates on a plane. This method of representation in a form \( r(\cos \theta + i \sin \theta) \), is intuitive for visualizing operations like rotation and scaling.

• \( r \) represents the modulus or magnitude, showing how "far" the number is from the origin on a plane.
• \( \theta \) signifies the angle, or direction, forming from the positive x-axis, indicating the direction or "rotation" the complex number points towards.

In the original exercise, \( r = -7 \) and \( \theta = 140^{\circ} \). Using these parameters, the number's position and magnitude got translated into rectangular form - a description relevant particularly for technical computing and visual modeling.
Polar coordinates find significant usage in fields such as engineering and physics due to their alignment with circular and periodic events.

Euler's Formula Demystified

Euler's formula, a key tool in the realm of complex numbers, bridges exponential functions with trigonometry through: \[ e^{i\theta} = \cos \theta + i \sin \theta \] This elegant relationship allows complex numbers expressed in polar form to be easily converted to exponential form, simplifying computations.
Euler's formula connects different ways to represent complex numbers:

• Circular Motion: Imagines a point moving on a circle, synthesizing the algebraic and geometric perspectives.
• Exponential Growth: Reflects patterns of exponential change in trigonometric terms.

In this exercise, Euler’s formula was crucial in translating from polar to rectangular form. By applying this formula, the original polar expression \( -7(\cos 140^{\circ} + i \sin 140^{\circ}) \) was reformulated and solved as a manageable exponential form, streamlining the calculation process.
The widespread implications of Euler's formula attest to its power, revolutionizing mathematical approaches across calculus, physics, and beyond.