Problem 52 - Precalculus

Question

Express each complex number in rectangular form. $$4\left[\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)\right]$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Trigonometric Expressions

The given complex number is expressed in polar form: \[ 4 \left(\cos\left(\frac{11 \pi}{6}\right) + i \sin\left(\frac{11 \pi}{6}\right)\right) \]Here, the modulus is \(4\), and the angle \( \theta \) is \( \frac{11\pi}{6} \). We need to compute both \( \cos\left(\frac{11 \pi}{6}\right) \) and \( \sin\left(\frac{11 \pi}{6}\right) \).

Step 2: Calculate the Cosine Component

Calculate \( \cos\left(\frac{11 \pi}{6}\right) \).Since \( \frac{11 \pi}{6} \) is in the fourth quadrant, cosine is positive. \[ \cos\left(\frac{11 \pi}{6}\right) = \cos\left(360^\circ - 30^\circ\right) = \cos\left(30^\circ\right) = \frac{\sqrt{3}}{2} \]

Step 3: Calculate the Sine Component

Calculate \( \sin\left(\frac{11 \pi}{6}\right) \). Since \( \frac{11 \pi}{6} \) is in the fourth quadrant, sine is negative.\[ \sin\left(\frac{11 \pi}{6}\right) = \sin\left(360^\circ - 30^\circ\right) = -\sin\left(30^\circ\right) = -\frac{1}{2} \]

Step 4: Substitute into the Rectangular Form

Substitute the values of the trigonometric functions into the formula to express the complex number in rectangular form: \[ 4 \left(\frac{\sqrt{3}}{2} + i\left(-\frac{1}{2}\right)\right) = 4\cdot \frac{\sqrt{3}}{2} + 4i \cdot \left(-\frac{1}{2}\right) \]

Step 5: Simplify the Expression

Simplify to find the rectangular form:Calculate the real part:\[ 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \]Calculate the imaginary part:\[ 4i \cdot \left(-\frac{1}{2}\right) = -2i \]Combine these to form the rectangular expression: \[ 2\sqrt{3} - 2i \]

Key Concepts

Rectangular Form

Complex numbers in rectangular form are expressed as a combination of a real part and an imaginary part. This is often written in the form \( a + bi \), where:

• \( a \) is the real part of the complex number.
• \( b \) is the coefficient of the imaginary part, \( i \), where \( i \) is the square root of \(-1\).

Expressing complex numbers in rectangular form is beneficial because it provides a straightforward way to perform arithmetic operations like addition and subtraction. For example, when a complex number is given in polar form, as \( r(\cos \theta + i \sin \theta) \), converting it to rectangular form involves calculating the cosine and sine to complete the expression.
In our exercise, we transformed \( 4 \left(\frac{\sqrt{3}}{2} + i \left(-\frac{1}{2}\right)\right) \) into the rectangular form: \( 2\sqrt{3} - 2i \). Here, \( 2\sqrt{3} \) is the real part, and \(-2i \) is the imaginary part.

Polar Coordinates

Polar coordinates provide a unique way to represent complex numbers. Instead of using the rectangular format \( a + bi \), the polar form uses the modulus and the angle from the positive x-axis. A complex number in polar form is expressed as \( r(\cos \theta + i \sin \theta) \).

• \( r \) is the modulus or the magnitude, representing the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument or angle, measured counterclockwise from the positive x-axis.

This form is particularly useful in multiplication, division, and finding powers or roots of complex numbers through De Moivre's Theorem.
In our example, the complex number \( 4 \left(\cos\left(\frac{11\pi}{6}\right) + i \sin\left(\frac{11\pi}{6}\right)\right) \) is given in polar coordinates with a modulus of 4 and an angle \( \frac{11\pi}{6} \). These values help us transition to the rectangular form.

Trigonometric Functions

Trigonometric functions, specifically cosine and sine, are essential when dealing with complex numbers in polar form. These functions help translate the angle \( \theta \) into a format useful for rectangular representation.

• Cosine (\( \cos \)) provides the horizontal component in the polar representation.
• Sine (\( \sin \)) provides the vertical component.

Complex numbers in polar form are expressed in terms of trigonometric functions as \( r(\cos \theta + i \sin \theta) \), which are sometimes noted as cis(\( \theta \)) for simplicity.
In our specific problem, the cosine of \( \frac{11\pi}{6} \) was calculated to be \( \frac{\sqrt{3}}{2} \) and the sine to be \(-\frac{1}{2} \). These calculations allowed us to express the polar form in the rectangular format \( 2\sqrt{3} - 2i \). Understanding how to find these trigonometric values is essential for working with complex numbers.