Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$\frac{5}{2}\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right]$$

Step-by-Step Solution

Answer

The complex number represented graphically will be a vector from the origin to a point \(2.5\) units away in the direction of \(-30^{\circ}\). The Cartesian form of the complex number is \(\frac{5\sqrt{3}}{4} - i\frac{5}{4}\).

Step 1: Locating the Complex Number in the Complex Plane

To locate the complex number in the complex plane, use the polar representation of the complex number, \(r[\cos(\theta) + i \sin(\theta)]\). The magnitude of the complex number is \(r=\frac{5}{2}\), and the angle is \(-30^{\circ}\). From the origin, move \(\frac{5}{2} = 2.5\) units in the direction of \(-30^{\circ}\), and mark the point.

Step 2: Converting the Complex Number to Cartesian Form

The Cartesian form of a complex number in the polar form is given as \(x = r \cos (\theta)\) and \(y = r \sin (\theta)\), where \(x\) is the real part and \(y\) is the imaginary part. Use these formulas, substituting \(r=\frac{5}{2}\) or \(2.5\) for \(r\) and \(-30^{\circ}\) for \(\theta\). Note that you have to convert \(-30^{\circ}\) to radians first, which gives \(\theta = -30 \times \frac {\pi}{180} = -\frac {\pi}{6}\) radians. Now, calculate \(x = 2.5 \cos (-\frac {\pi}{6})\) and \(y = 2.5 \sin (-\frac {\pi}{6})\), to get the Cartesian form of the complex number.

Step 3: Simplifying to get the Cartesian Form

For \(x = 2.5 \cos (-\frac {\pi}{6})\), cos of \(-\frac {\pi}{6}\) is the same as the cos of \(\frac {\pi}{6}\) due to the even-property of cosine. So, it gives \(x = 2.5 * \frac {\sqrt{3}}{2} = \frac {5\sqrt{3}}{4}\). For \(y = 2.5 \sin (-\frac {\pi}{6})\), the sine of \(-\frac {\pi}{6}\) is negative, so it yields \(y = 2.5 * -\frac {1}{2} = -\frac {5}{4}\). Hence, the Cartesian form of the complex number is \(\frac {5\sqrt{3}}{4} - i\frac {5}{4}\).

Key Concepts

Polar Form

Understanding the polar form of a complex number is essential when dealing with angular representation in the complex plane. This form expresses a complex number in terms of its magnitude and direction, which we often express as:

• Magnitude (\(r\)) - the distance from the origin (0,0) to the point on the complex plane, calculated as \(\sqrt{x^2 + y^2}\).
• Angle (\(\theta\)) - the angle the line makes with the positive x-axis, which is measured in radians or degrees.

In the given exercise, the polar form is \(\frac{5}{2}[\cos(-30^\circ) + i \sin(-30^\circ)]\). You can break this down as follows:

• The magnitude \(r = \frac{5}{2}\).
• The angle \(\theta = -30^\circ\), which is converted to radians: \(-\frac{\pi}{6}\).

Understanding this allows you to graphically locate the complex number on the plane by moving 2.5 units at an angle of \(-30^\circ\). This polar form is particularly useful for multiplication and division of complex numbers due to its reliance on trigonometric functions.

Cartesian Form

Once you have a complex number in polar form, converting it to Cartesian form is a straightforward process that involves trigonometric identities. In Cartesian form, a complex number is expressed as \(x + yi\), where:

• \(x\) is the real part, calculated as \(r \cos(\theta)\).
• \(y\) is the imaginary part, calculated as \(r \sin(\theta)\).

In the solution, you begin by determining the angle in radians, which is necessary for precise calculations. By converting \(-30^\circ\) to \(-\frac{\pi}{6}\) radians, you can accurately find:

• \(x = 2.5 \cos(-\frac{\pi}{6}) = \frac{5\sqrt{3}}{4}\).
  This follows because the cosine function is even, meaning \(\cos(-\theta) = \cos(\theta)\).
• \(y = 2.5 \sin(-\frac{\pi}{6}) = -\frac{5}{4}\).
  The sine function, by contrast, is odd, which implies \(\sin(-\theta) = -\sin(\theta)\).

Thus, the Cartesian form of the complex number is accurately represented as \(\frac{5\sqrt{3}}{4} - i \frac{5}{4}\). This form is practical for simpler addition and subtraction of complex numbers.

Complex Plane Representation

The complex plane offers a visual perspective for understanding complex numbers. It serves a similar purpose for complex numbers as the Cartesian plane does for real numbers. In this plane:

• The horizontal axis (x-axis) represents the real part of a complex number.
• The vertical axis (y-axis) denotes the imaginary part.

Consider the complex number from the exercise. To plot it, calculate its Cartesian components \(x = \frac{5\sqrt{3}}{4}\) and \(y = -\frac{5}{4}\). This results in plotting the point \(\left(\frac{5\sqrt{3}}{4}, -\frac{5}{4}\right)\) on the complex plane. By moving from the origin, 2.5 units towards opposite \(-30^\circ\):

• You find the precise location on the plane.
• This visualisation helps in understanding operations like addition, subtraction, and multiplication geometrically.

Thus, graphically representing complex numbers is a fundamental skill that helps bridge algebraic manipulation and geometric insight.