Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt{3}}{2} + i \frac{1}{2} \)

Step 1: Recognize the Expression

The expression given is a combination of the cosine and sine functions with the imaginary unit, written as \( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \). This format is typically associated with complex numbers in trigonometric form.

Step 2: Calculate Cosine Value

Evaluate the cosine function. Since \( \cos \frac{\pi}{6} \) corresponds to the angle of \( 30^\circ \), the exact value is \( \frac{\sqrt{3}}{2} \).

Step 3: Calculate Sine Value

Evaluate the sine function. Since \( \sin \frac{\pi}{6} \) corresponds to the angle of \( 30^\circ \), the exact value is \( \frac{1}{2} \).

Step 4: Write in Rectangular Form

A complex number written in rectangular form is represented as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Thus, \( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \) becomes \( \frac{\sqrt{3}}{2} + i \frac{1}{2} \).

Key Concepts

Rectangular Form

The rectangular form of a complex number is a way to express a complex number in the form of \(a + bi\). In this notation, \(a\) represents the real part of the complex number and \(b\) represents the imaginary part. This form is also called the Cartesian form, resembling how points are represented on the Cartesian plane.
- **Real Part**: This is the component on the horizontal axis in the complex plane. It denotes the "real" aspect without any imaginary component. - **Imaginary Part**: This lies on the vertical axis and is associated with the imaginary unit \(i\), completing the complex number.When you write a number like \(\frac{\sqrt{3}}{2} + i \frac{1}{2}\), you are already looking at its rectangular form. The number \(\frac{\sqrt{3}}{2}\) is the real part, while \(\frac{1}{2}\) is the coefficient of the imaginary part. Hence it means moving \(\frac{\sqrt{3}}{2}\) units along the real (horizontal) axis and \(\frac{1}{2}\) units upward on the imaginary (vertical) axis.

Trigonometric Form

The trigonometric form is another way of expressing complex numbers, often used to emphasize the geometric interpretation of the complex number. It is written as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus (or magnitude) of the complex number, and \(\theta\) is the argument (or angle).
- **Modulus \(r\)**: This is the distance from the origin to the point \((a, b)\) in the complex plane and is calculated using \(r = \sqrt{a^2 + b^2}\).- **Argument \(\theta\)**: This represents the angle formed with the positive direction of the real axis, measured in radians.In the exercise, \(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\) is already presented in trigonometric form, with an implicit magnitude \(r = 1\). The angle \(\frac{\pi}{6}\) indicates the direction in the complex plane, making it easy to switch to rectangular form if needed.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in the field of complex numbers. It is defined by the property \(i^2 = -1\). This unique definition allows \(i\) to be used in creating numbers that expand the real number line into a complex plane.
- **Property**: \(i\) is not a "real" number, which means it cannot be found on the real number line; instead, it extends the number system into two dimensions.- **Usage**: \(i\) is the backbone of the imaginary part of complex numbers, used to express rotations in the complex plane and to solve equations that do not have solutions in the set of real numbers.In expressions like \(\frac{1}{2}i\), \(i\) lets you understand and process the vertical displacement in the complex plane, adding a whole new dimension to number visualization.