Problem 50 - Precalculus

Question

Express each complex number in rectangular form. $$2\left[\cos \left(\frac{5 \pi}{6}\right)+i \sin \left(\frac{5 \pi}{6}\right)\right]$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Polar Form

The complex number is given in polar form as \( 2 \left[\cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right)\right] \), which is equivalent to \( 2 \cdot \text{cis} \left(\frac{5 \pi}{6}\right) \).

Step 2: Find the Cosine Value

Calculate \( \cos \left(\frac{5 \pi}{6}\right) \). The angle \( \frac{5 \pi}{6} \) radians (or 150 degrees) has a reference angle of 30 degrees in the second quadrant, where cosine is negative. Thus, \( \cos \left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2} \).

Step 3: Find the Sine Value

Calculate \( \sin \left(\frac{5 \pi}{6}\right) \). For \( \frac{5 \pi}{6} \) radians, the sine value is the same as \( \sin \left(\frac{\pi}{6}\right) \), which equals \( \frac{1}{2} \). Since it is in the second quadrant, sine remains positive: \( \sin \left(\frac{5 \pi}{6}\right) = \frac{1}{2} \).

Step 4: Convert to Rectangular Form

Substitute \( \cos \left(\frac{5 \pi}{6}\right) \) and \( \sin \left(\frac{5 \pi}{6}\right) \) back into the expression: \[ 2 \left(-\frac{\sqrt{3}}{2} + i \frac{1}{2}\right) \] Multiply through by 2: \[ -\sqrt{3} + i \] Thus, the rectangular form is \( -\sqrt{3} + i \).

Key Concepts

Rectangular Form

When we talk about the rectangular form of a complex number, we are dealing with how the number is represented in terms of its real and imaginary parts. Complex numbers merge real numbers and imaginary numbers into one entity. In mathematics, we often use the notation \( a + bi \) where:

• \( a \) is the real part
• \( b \) is the imaginary part

In this setup, the letter "i" represents the square root of -1.
In the exercise provided, we began with a complex number in polar form. Through calculation and substitution, it was converted into its rectangular form: \( -\sqrt{3} + i \). Here, \( -\sqrt{3} \) represents the real part, and \( 1 \), paired with "i", is the imaginary part. This form allows easy visualization of complex numbers as points in a plane.

Polar Coordinates

Polar coordinates offer us another powerful way to depict complex numbers. Unlike rectangular form, polar coordinates express a complex number in terms of a radius and an angle. The general polar form is written as \( r(\cos(\theta) + i\sin(\theta)) \), where:

• \( r \) is the magnitude (or modulus) of the vector
• \( \theta \) is the argument (or angle)

This format is particularly useful for multiplying and dividing complex numbers, as well as raising them to powers or extracting roots.
In our specific example, the initial representation was in polar form: \( 2(\cos \left(\frac{5 \pi}{6}\right) + i\sin \left(\frac{5 \pi}{6}\right)) \). It indicates the complex number with a magnitude of 2 and an angle of \( \frac{5\pi}{6} \) radians.

Trigonometric Functions

Trigonometric functions such as sine and cosine are pivotal when working with polar coordinates.

• \( \cos(\theta) \) provides the x-coordinate or the real part in the rectangular form.
• \( \sin(\theta) \) gives the y-coordinate or the imaginary part in the rectangular form.

The angle \( \frac{5 \pi}{6} \) used in our exercise corresponds to 150 degrees. It lies in the second quadrant of the unit circle on a Cartesian plane.
Here:

• The cosine of \( \frac{5 \pi}{6} \) is \(-\frac{\sqrt{3}}{2} \), reflecting its negative x-axis influence.
• Likewise, the sine value, \( \frac{1}{2} \), remains positive due to the positive y-axis influence.

Once these values are plugged into the original polar expression, they help convert the complex number into its rectangular form, highlighting how geometry and trigonometry blend seamlessly in complex number analysis.