Problem 55 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$3\left(\cos 100^{\circ}+i \sin 100^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(-0.5208 + 2.9544i\).

Step 1: Understand Polar Form

The complex number is given in polar form as \(3 ext{cis} heta\), where \(3\) is the modulus and \(\theta=100^{\circ}\) is the argument. The expression for polar form is: \( r\left(\cos\theta + i\sin\theta\right) \).

Step 2: Plug Values into the Formula

Substitute the values into the rectangular form equation based on the polar form equation: \(3(\cos 100^{\circ} + i \sin 100^{\circ})\).

Step 3: Calculate \(\cos 100^{\circ}\) and \(\sin 100^{\circ}\)

Use a calculator to find \(\cos 100^{\circ}\) and \(\sin 100^{\circ}\): \(\cos 100^{\circ} \approx -0.1736\) and \(\sin 100^{\circ} \approx 0.9848\).

Step 4: Multiply by the Modulus

Multiply both the cosine and sine components by the modulus: \[ 3 \times \cos 100^{\circ} = 3 \times (-0.1736) = -0.5208 \]\[ 3 \times \sin 100^{\circ} = 3 \times 0.9848 = 2.9544 \]

Step 5: Write the Complex Number in Rectangular Form

Combine the computed real and imaginary parts to write the number in rectangular form: \[-0.5208 + 2.9544i\].

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers that highlights their magnitude and direction. Instead of using real and imaginary components directly, it uses a modulus (or magnitude) and an angle (called the argument) to describe the number. This form is particularly useful because it reflects the geometric interpretation of complex numbers on the complex plane.

The general representation for polar form is given by:

• \( r(\cos \theta + i \sin \theta) \)
• Or more concisely as \( r \text{cis} \theta \)

Here, \(r\) is the modulus, and \(\theta\) is the argument of the complex number.
This method is heavily utilized in fields such as electrical engineering and physics because it makes multiplication and division of complex numbers very straightforward.

Complex Number

A complex number extends the idea of the real number system by incorporating imaginary numbers. It can be written in the form \(a + bi\), where:

• \(a\) is the real part and \(b\) is the imaginary part
• \(i\) is the imaginary unit, which satisfies \(i^2 = -1\)

These numbers are incredibly useful in various applications, ranging from engineering to quantum physics.

One of the unique aspects of complex numbers is their representation on the complex plane, which uses the real part for the x-axis and the imaginary part for the y-axis. This allows for operations like addition, subtraction, and even multiplication to be visualized geometrically.

Trigonometric Functions

Trigonometric functions play a crucial role in converting complex numbers from polar to rectangular form. They help determine the real and imaginary parts from a complex number given in polar form. Specifically:

• \(\cos\theta\): Determines the real part
• \(\sin\theta\): Determines the imaginary part

When you have a complex number in polar form, such as \(r(\cos\theta + i\sin\theta)\), calculating the cosine and sine of the argument \(\theta\) gives you the components needed to convert the number into rectangular form. This is why understanding trigonometric functions is essential when working with complex numbers in polar form.

Modulus and Argument

The modulus and argument of a complex number are key concepts when converting between polar and rectangular forms.

The **modulus** \(r\) of a complex number is the distance of the point from the origin in the complex plane. It is akin to the 'length' of the vector representing the complex number, calculated as \(r = \sqrt{a^2 + b^2}\) for a complex number \(a + bi\).
The **argument** \(\theta\) is the angle the vector forms with the positive x-axis, calculated using the arctan function: \(\theta = \arctan\left(\frac{b}{a}\right)\).

These two components allow us to effectively describe and manipulate complex numbers in polar form. They simplify not only basic arithmetic operations like multiplication and division but also advanced calculations like powers and roots.