Problem 26 - Algebra and Trigonometry

Question

Write each complex number in rectangular form. $$ 3\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right) $$

Step-by-Step Solution

Answer

-\frac{3\sqrt{3}}{2} - \frac{3}{2}i

Step 1 - Identify the given complex number in polar form

The given complex number is: \[ 3 \bigg( \text{cos} \frac{7 \pi}{6} + i \text{sin} \frac{7 \pi}{6} \bigg) \]

Step 2 - Find the cosine and sine values

\[ \text{cos} \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2} \] and \[ \text{sin} \frac{7 \pi}{6} = -\frac{1}{2} \]

Step 3 - Substitute the values back into the expression

Replace the cosine and sine values to get \[ 3 \bigg( -\frac{\sqrt{3}}{2} + i \bigg( -\frac{1}{2} \bigg) \bigg) \]

Step 4 - Simplify the multiplication

Distribute the 3: \[ 3 \times -\frac{\sqrt{3}}{2} + 3 \times -\frac{1}{2}i = -\frac{3\sqrt{3}}{2} - \frac{3}{2}i \]

Step 5 - Write the final rectangular form

Combine the real and imaginary parts: \[ -\frac{3\sqrt{3}}{2} - \frac{3}{2}i \]

Key Concepts

complex numbers

Complex numbers are numbers that extend the concept of one-dimensional numbers (like real numbers) to two dimensions. They have a real part and an imaginary part, which are usually represented as \(a + bi \). Here, \(a\) is the real part, and \(bi\) is the imaginary part where \(i\) is the imaginary unit and \(i^2 = -1 \). Complex numbers are incredibly useful in mathematics, engineering, and physics because they allow for the easy manipulation and representation of physical phenomena that cannot be represented with just real numbers.

polar to rectangular conversion

When a complex number is represented in polar form, it is expressed as \(r(\text{cos} \theta + i \text{sin} \theta)\). Here, \(r\) is the modulus (or magnitude) of the complex number, and \( \theta \) is the argument (or angle). To convert from polar to rectangular form, you use the cosine and sine values to find the real and imaginary parts:

• The real part is \( r \text{cos} \theta\)
• The imaginary part is \( r \text{sin} \theta\)

After converting, the rectangular form is simply the combination of these parts: \(a + bi\).

trigonometric form of complex numbers

The trigonometric form of a complex number is another way to represent complex numbers using an angle and a distance. It is given by \( r(\text{cos} \theta + i \text{sin} \theta) \), where:
- \(r\) is the modulus or magnitude
- \( \theta \) is the argument or angle
The trigonometric form simplifies multiplication and division of complex numbers because you can multiply the magnitudes and add the angles or divide the magnitudes and subtract the angles.

cosine and sine values

Finding the cosine and sine values is a crucial step in converting the trigonometric form to rectangular form. With respect to the exercise, we have:

• \( \text{cos} \frac{7 \frac{7 \text{\textbackslashpi}}{6} = -\frac{\textbackslashsqrt{3}}{2}\)
• \( \text{sin} \frac{7 \frac{7 \text{\textbackslashpi}}{6} = -\frac{1}{2}\)

Cosine and sine values can be found using the unit circle or trigonometric tables and must be substituted back into the original equation to assist with determining the rectangular form.

simplification of expressions

Simplification helps in making complex expressions easier to understand. In the exercise, once we have substituted the cosine and sine values into \(3 \bigg( -\frac{\text{\textbackslashsqrt{3}}}{2} + i \bigg( -\frac{1}{2} \bigg) \bigg)\), the next step is to perform the multiplication:
\( 3 \times -\frac{\text{\textbackslashsqrt{3}}}{2} = -\frac{3\text{\textbackslashsqrt{3}}}{2}\)
\( 3 \times -\frac{1}{2}i = -\frac{3}{2}i\)

Combining these, we get:
\( -\frac{3\text{\textbackslashsqrt{3}}}{2} - \frac{3}{2}i\)
Which is the simplified rectangular form of the original complex number in polar form.