Question

Find and plot the complex conjugate of each number. \(4\left(\cos \frac{2 \pi}{3}-i \sin \frac{2 \pi}{3}\right)\)

Short answer

The complex conjugate is \[ 4\bigg(\frac{-1}{2} + i \frac{\sqrt{3}}{2}\bigg) \].

Step-by-step solution

Identify the given complex number

The given complex number is expressed in polar form as: \[ 4\bigg(\text{cos} \frac{2\pi}{3} - i \text{sin} \frac{2\pi}{3}\bigg) \]

Recall the formula for the complex conjugate

The complex conjugate of a complex number in polar form \( r(\text{cos} \theta + i \text{sin} \theta) \) is \( r(\text{cos} \theta - i \text{sin} \theta) \).

Apply the conjugate formula

Applying the conjugate formula to the given complex number: \[ 4\bigg(\text{cos} \frac{2\pi}{3} - i(- \text{sin} \frac{2\pi}{3})\bigg) = 4\bigg(\text{cos} \frac{2\pi}{3} + i \text{sin} \frac{2\pi}{3}\bigg) \]

Simplify

The complex conjugate of the number is: \[ 4\bigg(\frac{-1}{2} + i \frac{\sqrt{3}}{2}\bigg) \]

Plot the Conjugate on the Complex Plane

To plot the complex conjugate, note the values: Real part: -2 Imaginary part: \[ 2\sqrt{3} \]. Plot the point \((-2, 2\sqrt{3})\) on the complex plane.

Key concepts

Polar Form

Complex numbers can be expressed in a polar form, which is especially useful for calculations involving multiplication and division. Polar form represents a complex number in terms of its magnitude (or modulus) and angle (or argument). This form is written as: \[ |r|(\text{cos} \theta + i \text{sin} \theta) \]. In the given exercise, the complex number is provided in polar form as: \[ 4\bigg(\text{cos} \frac{2\text{π}}{3} - i \text{sin} \frac{2\text{π}}{3}\bigg) \].To understand it:

• Magnitude (|r|) is 4
• Angle (θ) is \(\frac{2\text{π}}{3}\)

Complex Plane

The complex plane, or Argand plane, is a two-dimensional plane used to represent complex numbers graphically. The plane has a horizontal axis (real axis) and a vertical axis (imaginary axis). Each complex number corresponds to a unique point on the plane. For example, in the given exercise, the original complex number can be plotted using:

• Real part: -2
• Imaginary part: \(2\text{√}3\)

This helps clearly visualize the relationships and transformations, such as conjugation, of complex numbers.

Imaginary Part

The imaginary part of a complex number is the coefficient of the imaginary unit 'i'. In a complex number written as \(a + bi\), 'b' is the imaginary part. The imaginary part plays a crucial role in distinguishing complex numbers from real numbers. In the provided example: \[ 4\bigg(\text{cos} \frac{2\text{π}}{3} - i \text{sin} \frac{2\text{π}}{3}\bigg) \] simplifies to: \[ 4\bigg(\frac{-1}{2} - i \frac{\text{√}3}{2}\bigg) \].Here, \(- \frac{\text{√}3}{2}\) is the imaginary part multiplied by 4,resulting in:

• Imaginary part: -2\(\text{√}3\)

Real Part

The real part of a complex number is the component without the imaginary unit 'i'. In a complex number written as \(a + bi\), 'a' is the real part. The real part provides a point of reference on the real axis of the complex plane. For instance, in the example provided: \[ 4\bigg(\frac{-1}{2} - i \frac{\text{√}3}{2}\bigg) \], we get the real part to be:

• Real part: \(4 \times \frac{-1}{2} = -2\)

This helps significantly in understanding and plotting the complex number on the complex plane.

Step-by-Step Solution

A step-by-step solution method helps in understanding the process of finding the complex conjugate of a number. Breaking down the exercise:

**Step 1: Identify the given complex number**

Recognize the complex number in polar form: \[ 4\bigg(\text{cos} \frac{2\text{π}}{3} - i \text{sin} \frac{2\text{π}}{3}\bigg) \]

**Step 2: Recall the formula for the complex conjugate**

The formula is \[ r(\text{cos} \theta - i \text{sin} \theta) \] for polar forms.

**Step 3: Apply the conjugate formula**

Using the formula: \[ 4(\text{cos} \frac{2\text{π}}{3} + i \text{sin} \frac{2\text{π}}{3}) \]

**Step 4: Simplify**

Convert trigonometric values to real and imaginary parts: \[ 4\bigg(\frac{-1}{2} + i \frac{\text{√}3}{2}\bigg) \]

**Step 5: Plot the conjugate on the complex plane**

Plot the point

• Real part: -2
• Imaginary part: 2\( \text{√}3\)

Thus, the point \((-2,2 \text{√}3\)) can be plotted on the complex plane.