Problem 32 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(31-40\) , write the standard form of the complex number. Then represent the complex number graphically. $$5\left(\cos 135^{\circ}+i \sin 135^{\circ}\right)$$

Step-by-Step Solution

Answer

The standard form of the complex number \(5(\cos 135^{\circ}+i \sin 135^{\circ})\) is \(-\sqrt{2}/2 + \sqrt{2}i/2\). Graphically, it can be represented by the point \((- \sqrt{2}/2 , \sqrt{2}/2\)) on the complex plane.

Step 1: Convert Trigonometric Form to Standard Form

In the given complex number \(5(\cos 135^{\circ}+i \sin 135^{\circ})\), \(5\) is the magnitude (also called modulus) and \(135^{\circ}\) is the argument (or angle). To write this in standard form \(a + bi\), we can expand it as \(5\cos 135^{\circ} + 5i \sin 135^{\circ}\). Now calculate the values for cos 135^{\circ} and sin 135^{\circ} which are \(-\sqrt{2}/2\) and \(\sqrt{2}/2\) respectively. Therefore, the standard form becomes \(5*(-\sqrt{2}/2) + 5i*(\sqrt{2}/2)\) which simplifies as \(-5\sqrt{2}/2 + 5\sqrt{2}i/2\).

Step 2: Simplify the Standard Form

The complex number \(-5\sqrt{2}/2 + 5\sqrt{2}i/2\) can be simplified further. Both the real and imaginary parts share a common factor of \(5\sqrt{2} / 2\). So we can write the number as \(-\sqrt{2}/2 + \sqrt{2}i/2\).

Step 3: Graphic Representation

In order to graphically represent this complex number, we will plot it on a complex plane. A complex plane is similar to a Cartesian plane, and the horizontal axis represents the real part while the vertical axis represents the imaginary part. In this case, the real part is \(-\sqrt{2}/2\) and the imaginary part is \(\sqrt{2}/2\), so we will plot the point \((- \sqrt{2}/2 , \sqrt{2}/2\)) on the complex plane.

Key Concepts

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number is a compact way to represent complex numbers using the magnitude (modulus) and the argument (angle). It's based on the relationship between a point in a plane and its distance and direction from the origin. The trigonometric form is given by:
\[ r(\cos\theta + i\sin\theta) \]
Here, \( r \) is the magnitude of the complex number and \( \theta \) is the argument, measured in degrees or radians.
To convert it into the standard form \( a + bi \), we multiply \( r \) with the cosine and sine of the argument:
\[ a = r\cos\theta \] \[ b = r\sin\theta \]
In our example, the magnitude is \(5\) and the argument is \(135^\circ\), thus the conversion to standard form involves using trigonometric identities for \(135^\circ\). This aesthetically pleasing relationship marries algebra and geometry, and is especially handy in tasks involving multiplication or division of complex numbers.

Plotting Complex Numbers

Plotting a complex number on the complex plane is akin to pinpointing a location on a map. The complex plane is a two-dimensional plane with a horizontal axis (the real axis) and a vertical axis (the imaginary axis). Together, they help us visualize complex numbers spatially.
To plot a complex number, we treat the real part as the x-coordinate and the imaginary part as the y-coordinate. The point \( (a, b) \) on this plane represents the complex number \( a + bi \).
With our complex number \(-5\sqrt{2}/2 + 5\sqrt{2}i/2\), we identify the real part \(-5\sqrt{2}/2\) and the imaginary part \(5\sqrt{2}i/2\), leading us to the point \(( -\sqrt{2}/2, \sqrt{2}/2 )\) on the plane. This visualization is key in understanding the geometric interpretation of complex numbers and serves as a bridge to more intuitive learning.

Magnitude and Argument of Complex Numbers

Every complex number has two intrinsic features: magnitude and argument. The magnitude, also known as the modulus, is the distance from the origin to the point representing the complex number in the complex plane. It is calculated as:
\[ |z| = \sqrt{a^2 + b^2} \]
where \( a \) is the real part and \( b \) is the imaginary part of the complex number \( z = a + bi \).
The argument of a complex number is the angle that the line segment from the origin to the point makes with the positive real axis. It is denoted as \( \theta \) and can be calculated using the arctangent function:
\[ \theta = \arctan\left(\frac{b}{a}\right) \]
For our example, the magnitude is the coefficient \(5\), and the argument is \(135^\circ\). Understanding these concepts is essential for delving into complex analysis and for computations involving complex numbers, as they form the foundation for more advanced operations like exponential form and complex number calculus.