Problem 40 - 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. $$3\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The rectangular form is \(-\frac{3\sqrt{3}}{2} + \frac{3}{2}i\)."

Step 1: Identify the form and components

The given complex number is in polar form: \(r(\cos \theta + i \sin \theta)\), where \(r = 3\) and \(\theta = \frac{5 \pi}{6}\). Our goal is to convert this to rectangular form \(a + bi\).

Step 2: Calculate the cosine of the angle

We know \(\theta = \frac{5 \pi}{6}\). The cosine of \(\frac{5 \pi}{6}\) is known from trigonometry, which is \(\cos \frac{5 \pi}{6} = -\frac{\sqrt{3}}{2}\).

Step 3: Calculate the sine of the angle

Similarly, the sine of \(\frac{5 \pi}{6}\) is \(\sin \frac{5 \pi}{6} = \frac{1}{2}\).

Step 4: Compute the real part

Calculate the real part of the rectangular form: \(a = r \cos \theta = 3 \times -\frac{\sqrt{3}}{2} = -\frac{3\sqrt{3}}{2}\).

Step 5: Compute the imaginary part

Calculate the imaginary part of the rectangular form: \(b = r \sin \theta = 3 \times \frac{1}{2} = \frac{3}{2}\).

Step 6: Write the complex number in rectangular form

The complex number in rectangular form is given by combining the real and imaginary parts: \(a + bi = -\frac{3\sqrt{3}}{2} + \frac{3}{2}i\).

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers, which include an imaginary part. They are usually written as a combination of a real part and an imaginary part, in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\).

These numbers are incredibly useful in a wide array of fields, such as engineering, physics, and mathematics, due to their ability to model oscillations, rotations, and more.

To handle complex numbers effectively, it's important to understand both their rectangular form \(a + bi\) and their polar form, which can give insights into their magnitude and angle.

Polar Form Conversion

Polar form expresses a complex number in terms of a radius \(r\) (also known as the modulus) and an angle \(\theta\) (known as the argument). A complex number \(z\) in polar form is written as \(z = r(\cos \theta + i \sin \theta)\).

One of the main reasons to use polar form is its simplicity in performing multiplication and division of complex numbers, which can be reduced to simple operations of their magnitudes and arguments.

• The modulus \(r\) represents the distance from the origin to the point in the complex plane.
• The argument \(\theta\) is the angle formed with the positive x-axis.

To convert from polar to rectangular form, you multiply \(r\) by the cosine and sine of \(\theta\) to find the real and imaginary components, respectively.

Trigonometry

Trigonometry is essential in complex number operations, especially when dealing with polar form conversion. It involves understanding the relationships between the angles and sides of triangles, particularly right triangles.

In the context of complex numbers, trigonometric functions like cosine and sine are used to determine the real and imaginary parts of a complex number in polar form:

• \(\cos \theta\) gives the horizontal (real) component.
• \(\sin \theta\) gives the vertical (imaginary) component.

For the angle \(\theta = \frac{5\pi}{6}\), common trigonometric values are:
- \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\)
- \(\sin \frac{5\pi}{6} = \frac{1}{2}\).

These values help us convert the given complex number from polar to rectangular form.

Real and Imaginary Parts

The real and imaginary parts of a complex number are the fundamental components that define its position in the complex plane. When a complex number is given in rectangular form \(a + bi\),

• \(a\) is the real part.
• \(b\) is the imaginary part, multiplied by \(i\).

In the exercise, after the polar form conversion, the complex number \(3(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})\) becomes:
- Real part \(\left(a = 3 \times -\frac{\sqrt{3}}{2} = -\frac{3\sqrt{3}}{2}\right)\)
- Imaginary part \(\left(b = 3 \times \frac{1}{2} = \frac{3}{2}\right)\)

Combining these, we obtain the rectangular form: -\(\frac{3\sqrt{3}}{2} + \frac{3}{2}i\). Understanding these components allows for easy manipulation and combination of complex numbers.