Problem 6 - Complex numbers from A to...Z

Question

Find polar representations for the following complex numbers: a) \(z_{1}=\cos a-i \sin a, \quad a \in[0,2 \pi)\) b) \(z_{2}=\sin a+i(1+\cos a), \quad a \in[0,2 \pi)\) c) \(z_{3}=\cos a+\sin a+i(\sin a-\cos a), \quad a \in[0,2 \pi)\) d) \(z_{4}=1-\cos a+i \sin a, \quad a \in[0,2 \pi)\)

Step-by-Step Solution

Answer

Question: Find the polar representation for the following complex numbers: - \(z_1=\cos a-i\sin a\) - \(z_2=\sin a + i(1+\cos a)\) - \(z_3=\cos a + \sin a + i(\sin a - \cos a)\) - \(z_4=1 - \cos a + i\sin a\) Answer: The polar representations for the given complex numbers are: - \(z_1 = 1(\cos(-a) + i\sin(-a))\) - \(z_2 = \sqrt{2+2\cos a}(\cos(θ2) + i\sin(θ2))\) - \(z_3 = \sqrt{2}(\cos(θ3) + i\sin(θ3))\) - \(z_4 = \sqrt{2-2\cos a}(\cos(θ4) + i\sin(θ4))\)

Step 1: Find the modulus for each complex number

We need to calculate the modulus for each of the given complex numbers(z1, z2, z3, z4). Modulus formula: r = \(\sqrt{(\textit{real part})^2 + (\textit{imaginary part})^2}\)

Step 2: Find the argument for each complex number

Now, we have to find the argument or angle θ for each complex number. Argument formula: θ = \(\operatorname{atan2}(\textit{imaginary part}, \textit{real part})\)

Step 3: Write polar representation for each complex number

Write the polar representation for each complex number using the modulus and argument found in the previous steps. Polar representation formula: z = \(r(\operatorname{cos}\theta +i \operatorname{sin}\theta)\) Now we can proceed to solve individually:

Step 3a: Polar representation for z1

\(z_1=\cos a-i\sin a\) The modulus r1 is $ r_1=\sqrt{(\cos a)^2 + (-\sin a)^2} = \sqrt{1}=1.$ The argument θ1 can be found using atan2: θ1 = \(\operatorname{atan2}(-\sin a, \cos a) = -a\) (since, a ∈ [0, 2π)) The polar representation of z1 is: \(z_1 = 1(\cos(-a) + i\sin(-a))\)

Step 3b: Polar representation for z2

\(z_2=\sin a + i(1+\cos a)\) The modulus r2 is $ r_2=\sqrt{(\sin a)^2 + (1+\cos a)^2} = \sqrt{2+2\cos a}.$ The argument θ2 can be found using atan2: θ2 = \(\operatorname{atan2}(1 + \cos a, \sin a)\) The polar representation of z2 is: \(z_2 = \sqrt{2+2\cos a}(\cos(θ2) + i\sin(θ2))\)

Step 3c: Polar representation for z3

\(z_3=\cos a + \sin a + i(\sin a - \cos a)\) The modulus r3 is $ r_3=\sqrt{(\cos a+\sin a)^2 + (\sin a-\cos a)^2} = \sqrt{2}$ The argument θ3 can be found using atan2: θ3 = \(\operatorname{atan2}(\sin a - \cos a, \cos a + \sin a)\) The polar representation of z3 is: \(z_3 = \sqrt{2}(\cos(θ3) + i\sin(θ3))\)

Step 3d: Polar representation for z4

\(z_4=1 - \cos a + i\sin a\) The modulus r4 is $ r_4=\sqrt{(1-\cos a)^2 + (\sin a)^2} = \sqrt{2-2\cos a}$ The argument θ4 can be found using atan2: θ4 = \(\operatorname{atan2}(\sin a, 1 - \cos a)\) The polar representation of z4 is: \(z_4 = \sqrt{2-2\cos a}(\cos(θ4) + i\sin(θ4))\)

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually written in the form ```z = a + bi```where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(i^2 = -1\), which differentiates it from real numbers.

• The real part specifies the number on the horizontal axis (in the complex plane).
• The imaginary part specifies the number on the vertical axis.

Complex numbers are often represented geometrically as points or vectors in a plane known as the Argand plane. This geometric representation helps us visualize operations like addition and multiplication of complex numbers.

Modulus

The modulus of a complex number is a measure of its distance from the origin on the complex plane. It is calculated using the formula:```r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}```For example, for the complex number \(z_1 = \cos a - i \sin a\), the modulus is calculated as:```r_1 = \sqrt{(\cos a)^2 + (-\sin a)^2} = \sqrt{1} = 1```

• The modulus is always a non-negative number.
• It represents the "size" or "magnitude" of the complex number.

Understanding the modulus helps in identifying how far a complex point is from the origin, similar to the radial distance in polar coordinates.

Argument

The argument of a complex number is the angle the line representing the complex number makes with the positive real axis on the complex plane. It is calculated using the formula:```\theta = \operatorname{atan2}(\text{imaginary part}, \text{real part})```This formula uses the \(\text{atan2}\) function, which effectively computes the angle in the correct quadrant. For example, the argument \(\theta_1\) of the complex number \(z_1 = \cos a - i \sin a\) is:```\theta_1 = \operatorname{atan2}(-\sin a, \cos a) = -a```

• The argument usually lies between \(-\pi\) and \(\pi\) radians.
• The angle is measured in the counterclockwise direction from the positive x-axis.

The argument, combined with the modulus, provides a complete polar description of the complex number.

Trigonometric Form

The trigonometric form of a complex number combines its modulus and argument to represent it in polar coordinates. It is expressed as:```z = r(\cos \theta + i \sin \theta)```This form is particularly useful for multiplying or dividing complex numbers. It gives a clear view of both the magnitude and angle of the number.

• The modulus \(r\) indicates the distance from the origin.
• The angle \(\theta\) specifies the direction from the positive real axis.

For example, for \(z_1 = \cos a - i \sin a\), the trigonometric form is:```z_1 = 1 (\cos(-a) + i \sin(-a))```Using this form can simplify complex arithmetic, especially when performing powers and roots of complex numbers. The trigonometric form is fundamental when converting a complex number from rectangular (or Cartesian) coordinates to polar coordinates.