Problem 31 - Algebra and Trigonometry

Question

In Exercises \(27-36,\) write each complex number in rectangular form. If necessary, round to the nearest tenth. $$ 8\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular form of given complex number is \(4\sqrt{2} -4\sqrt{2} i\).

Step 1: Identify variables in the given equation

In the given complex number \(8\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)\), the number '8' is the radius \(r\), and \(\frac{7 \pi}{4}\) is the angle \(θ\) in radians.

Step 2: Calculate the real part \(a = r \cos θ\)

Using the radius and angle given, compute the real part of the complex number. The cosine of \(\frac{7 \pi}{4}\) is \(\sqrt{2}/2\). Therefore, \(a = r \cos θ = 8 \times \sqrt{2}/2 = 4\sqrt{2}\).

Step 3: Calculate the imaginary part \(b = r \sin θ\)

With the given radius and angle, compute the imaginary part of the complex number. The sine of \(\frac{7 \pi}{4}\) is \(-\sqrt{2}/2\). Therefore, \(b = r \sin θ = 8 \times (-\sqrt{2}/2) = -4\sqrt{2}\).

Step 4: Write the complex number in rectangular form

The rectangular form of the complex number is given by \(a + bi\). Substituting the calculated real and imaginary parts, the complex number in rectangular form is \(4\sqrt{2} -4\sqrt{2} i\).

Key Concepts

Rectangular Form

Complex numbers in rectangular form are expressed as a combination of a real part and an imaginary part. This can be written as \(a + bi\), where \(a\) represents the real component and \(b\) is the imaginary component. Here, \(i\) is the imaginary unit. Think of this form like the coordinates on a graph: \(a\) is the horizontal coordinate (often called the x-coordinate), and \(b\) is the vertical coordinate (often called the y-coordinate).

To convert a complex number from another form, such as polar form, to rectangular form, you will typically use trigonometric identities. For example, you use \(\cos \theta\) to find \(a\) and \(\sin \theta\) to find \(b\). Once computed, you combine these to express the complex number in the standard \(a + bi\) format.

• Real part (\(a\)): This is the component that doesn't involve \(i\).
• Imaginary part (\(b\)): This is the component that is multiplied by \(i\).

Polar Coordinates

Polar coordinates represent a complex number using a magnitude ( or 'radius') and an angle (\(\theta\)). This format can be likened to defining a point's location based on its direction and distance from the origin in a coordinate plane. A complex number in polar form is often written as \(r(\cos \theta + i \sin \theta)\).

In converting from polar to rectangular form, we use trigonometric functions:

• \(r\) determines the length of the line from the origin to the point. Think of it as how far away the number is from zero.
• \(\theta\) is the angle formed with the positive x-axis. In radians, this might appear as fractions involving \(\pi\).

By applying \(r \cos \theta\) and \(r \sin \theta\), we get the rectangular components \(a\) and \(b\), respectively.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\), which means \(i^2 = -1\). This peculiar property helps extend the real number system, allowing us to solve equations that don't have solutions among real numbers.

In the context of complex numbers, \(i\) represents the vertical component (or part) of the number when expressed in rectangular form. It is crucial to note that while \(i\) is an abstract concept, its mathematical utility lies in simplifying equations, particularly in electrical engineering and physics.

In rectangular form, a complex number will be written as \(a + bi\), explicitly using \(i\) to denote the imaginary component. This allows for clear differentiation between real and imaginary parts.

Trigonometric Form

Trigonometric form, also known sometimes as exponential or Euler’s form, offers another perspective on expressing complex numbers. This form is written as \(r(\cos \theta + i \sin \theta)\). The components of this form allow us to appreciate the geometry of complex numbers, focusing on their magnitude and direction.

In this form:

• \(r\) is known as the modulus or absolute value of the complex number, representing its distance from the origin in the complex plane.
• \(\theta\) is the argument or angle subtended by the line segment (representing the complex number) with the positive x-axis.

Trigonometric form is particularly useful in calculations involving multiplication and division of complex numbers, as operations are more straightforward when you handle them as magnitudes and angles rather than separately for their real and imaginary components.