Problem 34 - Algebra and Trigonometry

Question

In Exercises \(27-36,\) write each complex number in rectangular form. If necessary, round to the nearest tenth. $$ 7\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the given complex number is \(0 - 7i\).

Step 1: Identify the modulus and argument of the complex number

Given complex number in polar form is \(7(\cos \frac{3 \pi}{2} + i \sin \frac{3 \pi}{2})\). Here, the modulus \(r\) is 7 and the argument \(\theta\) is \(\frac{3 \pi}{2}\).

Step 2: Convert from polar to rectangular form

The real part \(x\) is given by \(r\cos \theta\) and the imaginary part \(y\) is given by \(r\sin \theta\). Therefore, calculate \(x\) and \(y\) using these formulas.

Step 3: Calculate real part of the complex number

The real part is \(x = r\cos \theta = 7\cos \frac{3 \pi}{2} = 0\).

Step 4: Calculate imaginary part of the complex number

The imaginary part is \(y = r\sin \theta = 7\sin \frac{3 \pi}{2} = -7\).

Step 5: Write the complex number in rectangular form

Substitute the values of \(x\) and \(y\) into the formula for the rectangular form of the complex number, \(x + yi\), to get the complex number in rectangular form.

Key Concepts

Polar Form of Complex Numbers

Understanding the polar form of complex numbers is essential in fields like engineering and physics, where rotational dynamics are analyzed. A complex number in polar form can be represented by the notation
\r\( r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) represents the modulus of the complex number, and \( \theta \) denotes its argument, or the angle it makes with the positive real axis. The modulus corresponds to the distance from the origin in the complex plane, while the argument provides the direction. It's a compact way to convey both magnitude and direction of a complex number.

Converting Polar to Rectangular Coordinates

When we convert a complex number from polar to rectangular coordinates, we are essentially translating circular movement into linear dimensions along the real and imaginary axes. This is done by using the cosine function for the real part, and the sine function for the imaginary part. The formulas for this are
\r\( x = r \cos \theta \) and \( y = r \sin \theta \), which give us the real and imaginary components, respectively. This process is crucial for simplifying the multiplication and division of complex numbers, as it is more straightforward in the polar form.

Complex Number Modulus and Argument

The modulus of a complex number, represented usually as \( |z| \) or simply \( r \), is its 'size' or 'length.' It's calculated as the square root of the sum of the squares of the real and imaginary parts, which is the distance from the origin to the point in the complex plane.

\rThe argument, denoted as \( \theta \), is the measure of the angle made with the positive real axis and can be found using trigonometry, specifically the tangent function, where \( \theta = \arctan(\tfrac{y}{x}) \) for a complex number \( x + yi \). The modulus and argument are vital for understanding the properties and behaviors of complex numbers.

Trigonometric Representation of Complex Numbers

Expressing complex numbers in trigonometric form involves utilizing sine and cosine functions to define their position on the complex plane. This representation leverages Euler's formula, which states that \( e^{i\theta} = \cos\theta + i\sin\theta \). Often used in electrical engineering and signal processing, this representation simplifies the process of raising complex numbers to powers or finding roots, as these operations become a matter of manipulating the modulus and adding or subtracting arguments.