Problem 10 - 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 \(\operatorname{Arg}(z)\). $$ z=2 \sqrt{2}-2 i \sqrt{2} $$

Step-by-Step Solution

Answer

The polar representation is \(z = 4(\cos(-\pi/4) + i \sin(-\pi/4))\), with \(\operatorname{Re}(z) = 2\sqrt{2}\), \(\operatorname{Im}(z) = -2\sqrt{2}\), \(|z| = 4\), and \(\arg(z) = \operatorname{Arg}(z) = -\pi/4\).

Step 1: Identify Real and Imaginary Parts

Firstly, recognize that the given complex number is in the form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) as the imaginary unit. In this case, \(z = 2\sqrt{2} - 2i\sqrt{2}\). Thus, the real part \(\operatorname{Re}(z)\) is \(2\sqrt{2}\), and the imaginary part \(\operatorname{Im}(z)\) is \(-2\sqrt{2}\).

Step 2: Calculate the Modulus

The modulus \(|z|\) of a complex number \(z = a + bi\) is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). By substituting \(a = 2\sqrt{2}\) and \(b = -2\sqrt{2}\), we obtain: \[ |z| = \sqrt{(2\sqrt{2})^2 + (-2\sqrt{2})^2} = \sqrt{8 + 8} = \sqrt{16} = 4. \]

Step 3: Find the Argument of z

The argument of a complex number \(z = a + bi\) can be found using \(\arg(z) = \arctan\left(\frac{b}{a}\right)\). Here, \(a = 2\sqrt{2}\) and \(b = -2\sqrt{2}\) which gives: \[ \arg(z) = \arctan\left(\frac{-2\sqrt{2}}{2\sqrt{2}}\right) = \arctan(-1). \] Since \(z\) is in the fourth quadrant, \(\arg(z) = -\frac{\pi}{4}.\)

Step 4: General Argument Determination

The principal value of the argument is \(\operatorname{Arg}(z)\), which is conventionally in the range \(-\pi < \operatorname{Arg}(z) \leq \pi\). Here, since \(-\frac{\pi}{4}\) already falls within this range, we have \(\operatorname{Arg}(z) = -\frac{\pi}{4}.\)

Step 5: Write the Polar Form

Finally, express the complex number in polar form. The polar form of a complex number is \(z = |z|(\cos(\theta) + i\sin(\theta))\). Using \(|z| = 4\) and \(\theta = -\frac{\pi}{4}\), the polar form is: \[ z = 4\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right). \]

Key Concepts

Complex Numbers

Complex numbers are fascinating numbers that include a real part and an imaginary part. The basic form of a complex number is expressed as \( z = a + bi \), where:\

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• \( a \) represents the real part,
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• \( b \) is the coefficient of the imaginary part, combined with the imaginary unit \( i \). \
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\The imaginary unit \( i \) fulfills the unique property \( i^2 = -1 \). Together, the real part and imaginary part create a new dimension for numerical calculations. This makes complex numbers incredibly useful in fields like engineering, physics, and applied mathematics. They allow us to solve problems that cannot be addressed with just real numbers alone. Understanding how complex numbers work forms the critical foundation for exploring their incredible applications.

Real and Imaginary Parts

The real and imaginary parts of a complex number are fundamental to understanding its full identity. These parts can be plainly seen in any complex number of the form \( z = a + bi \). \
In the expression \( z = 2\sqrt{2} - 2i\sqrt{2} \), to identify them easier, note that: \

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• The real part, \( \operatorname{Re}(z) \), is \( 2\sqrt{2} \). This is simply the coefficient sitting in front of the real part.
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• The imaginary part, \( \operatorname{Im}(z) \), is \( -2\sqrt{2} \). It includes both the coefficient \( -2\sqrt{2} \) and the imaginary unit \( i \).
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\Understanding these components helps one to graph the complex number on a two-dimensional plane called the complex plane. The real part is plotted on the horizontal axis, while the imaginary part is on the vertical axis.

Modulus of Complex Number

The modulus of a complex number offers insight into its size or magnitude. \
It's a measure of how far the number is from the origin of the complex plane, calculated with the formula:

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• \(|z| = \sqrt{a^2 + b^2} \)

For the complex number \( z = 2\sqrt{2} - 2i\sqrt{2} \), the real part \( a \) is \( 2\sqrt{2} \) and the imaginary part \( b \) is \( -2\sqrt{2} \). Plugging these into the formula gives us:
\[|z| = \sqrt{(2\sqrt{2})^2 + (-2\sqrt{2})^2} = \sqrt{8 + 8} = \sqrt{16} = 4.\]
The modulus is a key component of representing complex numbers in polar form, helping to define not just the distance but also to bridge the connection with the argument angle.

Argument of Complex Number

The argument of a complex number reveals the angle \( \theta \) it forms with the positive real axis on the complex plane. \
It's typically measured in radians and helps complete the polar representation. For any complex number \( z = a + bi \), the argument is found using the formula\( \arg(z) = \arctan\left(\frac{b}{a}\right) \).
With the example \( z = 2\sqrt{2} - 2i\sqrt{2} \), the argument calculation becomes:

• \( \arg(z) = \arctan\left(\frac{-2\sqrt{2}}{2\sqrt{2}}\right) = \arctan(-1) \)

Since the complex number lies in the fourth quadrant of the plane, the argument is \(-\frac{\pi}{4} \). The argument can also be expressed in terms of its principal value using \( \operatorname{Arg}(z) \), which remains within the range \(-\pi < \operatorname{Arg}(z) \leq \pi \). For \( z \), it's already within this range: \( \operatorname{Arg}(z) = -\frac{\pi}{4} \). Understanding the argument allows for a full representation of the complex number's position and angle, crucial in fields requiring precise geometric and trigonometric calculations.