Question

Find and plot the complex conjugate of each number. $\cos \pi -i\sin \pi$

Short answer

The complex conjugate is $-1$.

Step-by-step solution

Identify the complex number

The complex number given is $\text{cos}(\pi )-i\text{sin}(\pi )$.

Use the conjugate property

The complex conjugate of a complex number $a+bi$ is obtained by changing the sign of the imaginary part. For $a-bi$, the complex conjugate is $a+bi$.

Apply the conjugate property

The given complex number is $\text{cos}(\pi )-i\text{sin}(\pi )$. Its complex conjugate will be $\text{cos}(\pi )+i\text{sin}(\pi )$.

Calculate the values of cosine and sine at $\pi$

We know that $\text{cos}(\pi )=-1$ and $\text{sin}(\pi )=0$.

Substitute the values

Substituting the values found in Step 4, the conjugate becomes $-1+i\cdot 0$, which simplifies to $-1$.

Plot the complex conjugate

The complex number $-1$ is a real number and can be plotted on the real axis at $-1$.

Key concepts

complex number

A complex number is a combination of a real number and an imaginary number. It is usually written in the form $a+bi$, where $a$ represents the real part, and $bi$ represents the imaginary part. The imaginary unit, denoted by $i$, is defined as the square root of -1. Complex numbers can represent many concepts in mathematics and engineering. For example, if we have $3+4i$, 3 is the real part and $4i$ is the imaginary part.

conjugate property

The complex conjugate of a complex number changes the sign of the imaginary part. If a complex number is $a+bi$, its conjugate is $a-bi$. This property is useful for simplifying mathematical operations involving complex numbers. Conjugates are often used in dividing complex numbers or in other algebraic manipulations. For instance, the conjugate of $3+4i$ would be $3-4i$.

plotting complex numbers

Plotting complex numbers involves placing them in a two-dimensional plane, known as the complex plane or Argand diagram. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. For example, to plot the complex number $3+4i$, you would move 3 units along the real axis and then 4 units up the imaginary axis. The point where you end up is where you plot the complex number. For real numbers like $-1$, they are simply plotted on the real axis.

trigonometric functions

Trigonometric functions, such as cosine and sine, relate to the angles and sides of triangles. In the context of complex numbers, they are often used to express complex numbers in polar form. For example, a complex number $z=r(\text{cos}(\theta )+i\text{sin}(\theta ))$ describes a point with magnitude $r$ and angle $\theta$. Euler's formula $e^{i\theta }=\text{cos}(\theta )+i\text{sin}(\theta )$ is a powerful tool that links trigonometry and complex numbers together. Evaluating these functions at specific angles helps solve many mathematical problems.

imaginary unit

The imaginary unit, denoted by $i$, is a fundamental concept in complex numbers, defined as $i=\sqrt{-1}$. This leads to the properties $i^2=-1$, $i^3=-i$, and $i^4=1$. It allows for the extension of the real number system to include solutions to equations like $x^2=-1$. When working with complex numbers, $i$ plays a crucial role in forming the imaginary part. For example, in the complex number $3+4i$, $4i$ is the imaginary part, and it indicates a magnitude of 4 in the direction of the imaginary axis.