Question

Find and plot the complex conjugate of each number. $$ 7\left(\cos 110^{\circ}-i \sin 110^{\circ}\right) $$

Short answer

The complex conjugate of the given number is \[7 \left( \cos 110^{\circ} + i \sin 110^{\circ} \right)\] and it is plotted in the third quadrant.

Step-by-step solution

Review the Given Complex Number

The given complex number is represented in polar form: \[7\left(\cos 110^{\circ} - i \sin 110^{\circ}\right)\]Here, the modulus is 7 and the argument is 110°.

Understand the Concept of Complex Conjugate

The complex conjugate of a complex number in the form of \[a + bi\] is \[a - bi\].For a complex number in polar form \[r(\cos \theta + i \sin \theta)\], its complex conjugate is \[r(\cos \theta - i \sin \theta)\].

Apply the Concept to the Given Number

For the given number \[7\left(\cos 110^{\circ} - i \sin 110^{\circ}\right)\], its conjugate is \[7\left(\cos 110^{\circ} + i \sin 110^{\circ}\right).\]

Plot the Original and Conjugate Numbers

To plot these numbers on the complex plane, use the magnitude as the radius and the argument as the angle in the polar coordinate system:- The original number with argument 110° lies in the second quadrant.- The conjugate number with argument -110° (same magnitude) lies in the third quadrant. Plot points and draw vectors from the origin to these points.

Key concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
For example, \(3 + 4i\) is a complex number with a real part of 3 and an imaginary part of 4i.
Imaginary numbers are based on \(i\), which is defined as the square root of -1: \(i = \sqrt{-1}\).
This means that \(i^2 = -1\).

Polar Coordinates

Polar coordinates provide a way to represent complex numbers using a radius and an angle instead of a real and imaginary part.
In polar form, a complex number is written as \((r, \theta)\), where \(r\) is the modulus (or magnitude) and \(\theta\) is the argument (or angle).
The radius \(r\) is the distance from the origin to the point in the complex plane, and the angle \(\theta\) is measured from the positive real axis.
For example, a complex number can be converted from polar form to rectangular form (i.e., \(a + bi\)) using the following formulas:

• a = r \cos(\theta)\
• bi = r \sin(\theta)\

This way, the polar form \(r(cos(\theta) + i sin(\theta))\) corresponds to the rectangular form \(a + bi\).

Argument of a Complex Number

The argument of a complex number is the angle \(\theta\) formed between the positive real axis and the line representing the complex number in the complex plane.
It can be found using the inverse tangent function:
\[\theta = \operatorname{tan}^{-1}\left(\frac{b}{a}\right)\]
where \(a\) and \(b\) are the real and imaginary parts of the complex number, respectively.
The argument can take on values between \(-\pi\) and \(\pi\) (in radians) or \(-180^{\circ}\) and \(180^{\circ}\) (in degrees).
For example, for the complex number \(3 + 4i\), the argument is \(\operatorname{tan}^{-1}\left(\frac{4}{3}\right) = 53.13^{\circ}\).

Modulus of a Complex Number

The modulus (or magnitude) of a complex number is the distance from the origin to the point representing the complex number in the complex plane.
It can be calculated using the Pythagorean theorem:
\[r = \sqrt{a^2 + b^2}\]
where \(a\) and \(b\) are the real and imaginary parts of the complex number.
For the complex number \(3 + 4i\), the modulus is:
\[r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
This value represents the 'length' of the complex number from the origin in the complex plane.