Problem 7 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-\frac{\sqrt{3}}{2}-\frac{1}{2} i $$

Step-by-Step Solution

Answer

The polar form is \( z = 1( \cos\frac{7\pi}{6} + i\sin\frac{7\pi}{6} ) \); \( \operatorname{Re}(z) = -\frac{\sqrt{3}}{2}, \operatorname{Im}(z) = -\frac{1}{2}, |z| = 1, \arg(z) = \frac{7\pi}{6}, \operatorname{Arg}(z) = -\frac{\pi}{6} \).

Step 1: Identify Real and Imaginary Parts

The complex number is given as \( z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \). The real part, \( \operatorname{Re}(z) \), is \(-\frac{\sqrt{3}}{2}\) and the imaginary part, \( \operatorname{Im}(z) \), is \(-\frac{1}{2}\).

Step 2: Calculate Magnitude (Modulus)

The magnitude \(|z|\) of a complex number \( z = a + bi \) is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \). Here, \( a = -\frac{\sqrt{3}}{2} \) and \( b = -\frac{1}{2} \). Thus, \(|z| = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1\).

Step 3: Determine Argument (Phase)

The argument of a complex number is given by \( \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \). However, due to the complex number being in the third quadrant, adjustments are needed. First calculate \( \arg(z) = \tan^{-1}\left( \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} \right) = \frac{1}{\sqrt{3}}\) giving us an angle of \(\frac{\pi}{6}\). Since \(z\) is in the third quadrant, the angle should be \( \pi + \frac{\pi}{6} = \frac{7\pi}{6} \).

Step 4: Obtain Principal Argument

The principal argument \( \operatorname{Arg}(z) \) is the value of \( \arg(z) \) within the range \( (-\pi, \pi] \). Given that \( \frac{7\pi}{6} = 1.16666\pi\) which is more than \(\pi\), we must consider \( -\frac{\pi}{6} \) as the principle argument since they are coterminal.

Step 5: Write the Polar Form

The polar form of a complex number is given by \( z = |z|(\cos\theta + i\sin\theta) \). So we write the polar form as \( z = 1 \left( \cos \frac{7\pi}{6} + i\sin \frac{7\pi}{6} \right) \).

Key Concepts

Real and Imaginary Parts of Complex Numbers

When dealing with complex numbers, it is vital to understand that they are composed of two main parts: the real part and the imaginary part. Any complex number can be expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is known as the real part, represented as \( \operatorname{Re}(z) \), and \( b \) is the imaginary part, represented as \( \operatorname{Im}(z) \). In the given exercise, the complex number is \( z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \). This means \( \operatorname{Re}(z) = -\frac{\sqrt{3}}{2} \), and \( \operatorname{Im}(z) = -\frac{1}{2} \). Recognizing these parts is crucial when performing operations on complex numbers or converting them to polar form.

Magnitude of a Complex Number

The magnitude of a complex number, often called the modulus, measures the 'size' or 'length' of the complex number from the origin in the complex plane. It is symbolized by \( |z| \) and calculated using the Pythagorean theorem. For a complex number \( z = a + bi \), the magnitude is given by the formula: \( |z| = \sqrt{a^2 + b^2} \). In the exercise, where \( z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \), we find:

• \( a = -\frac{\sqrt{3}}{2} \)
• \( b = -\frac{1}{2} \)

Substitute these values into the formula to get \[ |z| = \sqrt{ \left( -\frac{\sqrt{3}}{2} \right)^2 + \left( -\frac{1}{2} \right)^2 } = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]. This shows that the magnitude of the given complex number is 1.

Argument of a Complex Number

The argument of a complex number, denoted as \( \arg(z) \), describes the angle a complex number forms with the positive real axis in the complex plane. The formula to calculate this angle is \( \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \). However, it is essential to pay attention to the quadrant where the complex number lies to determine the correct angle. The complex number \( z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \) is located in the third quadrant. First, calculating using the formula for \( \arg(z) \), we have:

• \( \arg(z) = \tan^{-1}\left(\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \)
• This gives an initial angle of \( \frac{\pi}{6} \)
• In the third quadrant, we adjust by adding \( \pi \) to get \( \frac{7\pi}{6} \)

This adjustment is crucial to ensure the angle accurately reflects the complex number's position in the plane.

Principal Argument

The principal argument, \( \operatorname{Arg}(z) \), represents the unique value for the argument of a complex number within the range \( (-\pi, \pi] \). This means no matter where the initial angle lies after considering the quadrant, the principal argument should always be adjusted to fit this range. For the complex number \( z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \), the calculated argument \( \arg(z) = \frac{7\pi}{6} \) exceeds \( \pi \). Since angles \( \frac{7\pi}{6} \) and \( -\frac{\pi}{6} \) are coterminal:

• Both these angles lie on the same direction on the plane but differ in numerical representation
• According to the principal argument range, we select \( -\frac{\pi}{6} \) as \( \operatorname{Arg}(z) \)

This ensures the argument is packaged within the standard range consistently applied to all complex numbers.