Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$1.5\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The standard form of the complex number is \(1.5i\). Plot this as a point on the Argand Plane at y=1.5 (the imaginary axis), x=0 (the real axis).

Step 1: Understand the Polar Form

Establish what is provided through the given polar form, which in this case is \(1.5(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2})\). This means the modulus (or distance from the origin) is 1.5, and the angle (argument), represented in radians, is \(\frac{\pi}{2}\).

Step 2: Translate to Standard Form

Using the trigonometric identities \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\), the polar form can be translated into standard form. Therefore, \(1.5(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2})\) translates to \(1.5 * 0 + 1.5*1*i = 0 + 1.5i\), which simplifies to \(1.5i\).

Step 3: Graphical Presentation

To graphically represent a complex number in the Argand Plane, plot a point where the x-coordinate is the real component (which is 0 in this case), and the y-coordinate is the imaginary part (which is 1.5 in this case).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in multiple forms, with the polar form being particularly useful for multiplication, division, and power operations. In the polar form, a complex number is expressed in terms of its magnitude and direction relative to the origin on a plane. This is akin to specifying how far and in what direction to travel from a central point.

The polar form is written as \( r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) is the modulus representing the distance from the origin, and \( \theta \) is the argument representing the angle with the positive real axis. It's a compact way to handle complex numbers in trigonometric contexts and for translating them into the standard form.

Converting Polar to Standard Form

Transitioning from the polar form to the standard form is a straightforward process, with the primary goal being to express the complex number as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. By applying the fundamental trigonometric identities such as \( \text{cos} \theta = \text{Re}/r \) and \( \text{sin} \theta = \text{Im}/r \), where \( \text{Re} \) and \( \text{Im} \) denote the real and imaginary parts respectively, one can convert \( r(\text{cos} \theta + i\text{sin} \theta) \) into \( a + bi \) form effortlessly.

It's crucial to recognize that the cosine and sine functions describe the coordinates of a point on the unit circle correlating with the angle \( \theta \). This principle allows for the decoding of the polar expression into a Cartesian point.

Graphing Complex Numbers

The graphical representation of complex numbers is performed on the complex plane, also known as the Argand plane, where the horizontal axis (x-axis) represents the real part of the complex number, and the vertical axis (y-axis) represents the imaginary part. To graphically plot a complex number, one can consider it analogous to plotting coordinates on a standard Cartesian plane.

Each complex number corresponds to a unique point in the plane. For instance, a number in standard form \( a + bi \) would be located at the point \( (a, b) \). In the context of the given exercise, the complex number in polar form requires no real part, as it is mapped directly upwards on the imaginary axis at a distance determined by its modulus.

Trigonometric Identities

A strong grasp of trigonometric identities is essential when working with complex numbers in polar form. These identities, such as \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \) and the angle sum formulas, allow for the simplification and transformation of trigonometric expressions within complex numbers.

Trigonometric identities express the relationships between the angles and sides of a triangle, and by extension, the coordinates of points on a unit circle. When converting complex numbers from polar to standard form, knowing that \( \text{cos} \frac{\pi}{2} = 0 \) and \( \text{sin} \frac{\pi}{2} = 1 \) negates the need for extensive calculation, instantly revealing that the corresponding Cartesian coordinates are exclusively on the imaginary axis.