Question

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

Short answer

The complex conjugate is \(\text{z}^* = i\).

Step-by-step solution

Identify the Given Complex Number

Given the complex number is \(\text{z} = \cos \left( \frac{3 \pi}{2} \right) + i \sin \left( \frac{3 \pi}{2} \right)\).

Evaluate the Trigonometric Functions

Find \(\text{cos} \left( \frac{3 \pi}{2} \right)\) and \(\text{sin} \left( \frac{3 \pi}{2} \right)\). \(\text{cos} \left( \frac{3 \pi}{2} \right) = 0\) and \(\text{sin} \left( \frac{3 \pi}{2} \right) = -1\). Therefore, \(\text{z} = 0 - i\).

Find the Complex Conjugate

The complex conjugate of \(\text{z} = a + bi\) is \(\text{z}^* = a - bi\). Hence, for \(\text{z} = -i\), the complex conjugate is \(\text{z}^* = i\).

Plot the Numbers

On the complex plane, represent \(\text{z} = -i\) at the point (0, -1) and \(\text{z}^* = i\) at the point (0, 1).

Key concepts

complex number

A complex number combines a real number and an imaginary number. It's written in the form \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part. The imaginary unit \(i\) is defined as the square root of -1. This leads to the fundamental property \(i^2 = -1\). Complex numbers are essential in various fields including engineering, physics, and applied mathematics. They help describe phenomena that are not easily represented by real numbers alone.

trigonometric functions

Trigonometric functions link angles of a triangle to the ratios of its sides and are essential in complex number calculations. Common trigonometric functions include cosine (\(\cos\)) and sine (\(\sin\)). For example, for an angle \(\theta = \frac{3\pi}{2}\), we use:

• \(\cos \left( \frac{3\pi}{2} \right) = 0\)
• \(\sin \left( \frac{3\pi}{2} \right) = -1\)

These functions simplify the representation of complex numbers and are crucial for converting them into different forms such as Euler's formula.

complex plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to represent complex numbers graphically. Each point on the complex plane corresponds to a unique complex number \(a + bi\). The horizontal axis (real axis) represents the real part \(a\), and the vertical axis (imaginary axis) represents the imaginary part \(bi\). This visual representation simplifies understanding and manipulating complex numbers, including operations like addition, subtraction, and finding conjugates.

plotting complex numbers

Plotting complex numbers involves representing them as points on the complex plane. For instance, the complex number \( -i\), where the real part is 0 and the imaginary part is -1, is plotted at the point (0, -1). Its conjugate, \( i\), is plotted at the point (0, 1). This graphical method helps in visualizing operations on complex numbers. For example, when you plot \(z = -i\) and its conjugate \(z^* = i\), you clearly see their symmetric positions relative to the real axis. This symmetry is crucial in complex number operations and their geometrical interpretations.