Problem 25 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ 2-i \sqrt{3} $$

Step-by-Step Solution

Answer

The complex number \(2-i\sqrt{3}\) is located in the fourth quadrant of the complex plane. Its polar form is \(2(\cos(-60^\circ) + i \sin(-60^\circ))\) or \(2(\cos(-\pi/3) + i \sin(-\pi/3))\) in radians.

Step 1: Plot the Complex Number

The complex number \(2-i\sqrt{3}\) is a point in the complex plane, where the real part (x-coordinate) is 2, and the imaginary part (y-coordinate) is \(-\sqrt{3}\). The imaginary part is negative, which means this point is in the fourth quadrant of the complex plane.

Step 2: Convert to Polar Form

A complex number can be represented in polar form as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument of the complex number. The modulus \(r\) is given by \(\sqrt{x^2 + y^2}\), and the argument \(\theta\) is given by \(\tan^{-1}(y/x)\). For \(2-i\sqrt{3}\), \(r = \sqrt{2^2 + (-\sqrt{3})^2} = 2\), and \(\theta = \tan^{-1}((-\sqrt{3})/2) = -60 ^\circ\) or \(-\pi/3\) rad, considering the complex number is in the fourth quadrant.

Step 3: Write the Complex Number in Polar Form

Finally, the polar form of the complex number \(2-i\sqrt{3}\) is given by \(r(\cos \theta + i \sin \theta)\), so \(2-i\sqrt{3} = 2(\cos(-60^\circ) + i \sin(-60^\circ))\) or \(2(\cos(-\pi/3) + i \sin(-\pi/3))\) in radians. Hence, both forms represent the same complex number.

Key Concepts

Polar Form

The polar form of a complex number is a way of expressing complex numbers by focusing on their magnitude and direction rather than their position on the real and imaginary axis. A complex number like the one given in the exercise, \(2-i\sqrt{3}\), can be converted into polar form by identifying its modulus and argument. The polar form is written as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the modulus, or absolute value, and \(\theta\) is the argument, or angle.

• The modulus \(r\) represents the distance from the origin to the point in the complex plane.
• The argument \(\theta\) describes the angle formed with the positive real axis.

The formula connects complex numbers to trigonometry, simplifying tasks like multiplication and division. For example, when multiplying complex numbers, you can easily multiply their moduli and add their arguments. The polar form becomes particularly useful in advanced mathematical operations or when dealing with oscillations in engineering.

Complex Plane

The complex plane is a two-dimensional plane that represents complex numbers graphically. You can visualize each complex number as a point or a vector on this plane. The layout is similar to a coordinate plane, where:

• The horizontal axis (real axis) represents the real part of the complex number.
• The vertical axis (imaginary axis) represents the imaginary part.

For the complex number \(2 - i\sqrt{3}\), \(2\) is the real part and \(-\sqrt{3}\) is the imaginary part. This will position the point in the fourth quadrant of the plane because the real part is positive and the imaginary part is negative. This visual representation helps in understanding relationships between different complex numbers and is particularly useful in solving and verifying complex number problems.

Modulus

The modulus of a complex number is its distance from the origin on the complex plane. It acts similarly to the radius in polar coordinates. Calculating the modulus involves using the Pythagorean theorem, where the modulus \(r\) is found by taking the square root of the sum of the squares of the real and imaginary parts.For example, for the complex number \(2 - i\sqrt{3}\), the modulus is computed as:\[ r = \sqrt{2^2 + (-\sqrt{3})^2} = \sqrt{4 + 3} = \sqrt{7} = 2\]This real value indicates the distance of the complex number from the origin, and is crucial for converting between rectangular and polar forms. The modulus also plays a pivotal role in determining the magnitude of vectors and signals represented by complex numbers.

Argument

The argument of a complex number is the angle the vector (representing the complex number) makes with the positive real axis. It is expressed in either degrees or radians. Finding the argument allows us to understand the rotational aspect of a complex number in the polar coordinate system.To determine the argument \(\theta\), you use the arctangent function:\[ \theta = \tan^{-1}(\frac{y}{x})\]where \(x\) is the real part and \(y\) is the imaginary part of the complex number. For example, for \(2 - i\sqrt{3}\), the argument is:\[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{2}\right) = -60^\circ \quad \text{or} \quad -\frac{\pi}{3}\, \text{radians}\]It's important to take into account the quadrant of the complex number to ensure the angle agrees with the standard position rules on the complex plane. The argument is crucial for converting a complex number to its polar form, as it helps in understanding its orientation.