Problem 29 - Precalculus with Limits

Question

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(5 + 2i\)

Step-by-Step Solution

Answer

The complex number \(5 + 2i\) is represented graphically as the point (5, 2) in the Cartesian plane. Its trigonometric form is \(\sqrt{29}(\cos(arctan(\frac{2}{5})) + i \sin(arctan(\frac{2}{5})))\).

Step 1: Plot the Complex Number

We have a complex number \(5 + 2i\). On the complex plane (also known as the Argand plane), the real part (5) is plotted along the x-axis, and the imaginary part (2) is plotted along the y-axis. So, this complex number represents the point (5, 2) in the Cartesian plane.

Step 2: Calculate the Magnitude

The magnitude of a complex number is calculated as \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the complex number respectively. Substituting \(a = 5\) and \(b = 2\), we get \(r = \sqrt{5^2 + 2^2} = \sqrt{29}\).

Step 3: Calculate the Angle

The angle or argument of a complex number is obtained as \(\theta = arctan(\frac{b}{a})\), where \(b\) and \(a\) are the imaginary and real parts of the complex number respectively. By substituting \(a = 5\) and \(b = 2\), we get \(\theta = arctan(\frac{2}{5})\).

Step 4: Write the Trigonometric form

Now that we have both the magnitude and the angle, we can write the trigonometric form of the complex number as \(r(\cos\theta + i \sin\theta)\). Substituting \(r = \sqrt{29}\) and \(\theta = arctan(\frac{2}{5})\), we get the trigonometric form as \(\sqrt{29}(\cos(arctan(\frac{2}{5})) + i \sin(arctan(\frac{2}{5})))\).

Key Concepts

Complex Plane Representation

One of the cornerstones of understanding complex numbers is visualizing them on the complex plane. Imagine a graph where instead of just the familiar x and y axes of algebra, we have an additional axis representing the imaginary numbers. The horizontal axis (x-axis) represents the real part of the complex number, while the vertical axis (y-axis) holds the imaginary part.

The graphical depiction of the complex number 5 + 2i becomes quite intuitive with this approach. Plotting this number involves locating the point where the real part (5) aligns on the x-axis and the imaginary part (2) along the y-axis. Collectively, this point corresponds to the coordinate (5, 2) on the complex plane. This is not just a geometrical exercise; plotting complex numbers allows us to easily determine other properties such as magnitude and argument, which are crucial in translating the number into its trigonometric form.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number is essentially its 'size' or 'length' when we consider the complex number as representing a point in the complex plane. Just as we can measure the distance of a straight line in a two-dimensional space using the Pythagorean theorem, we similarly calculate the magnitude of a complex number.

For our example of the complex number 5 + 2i, the magnitude is found using the formula: \(r = \sqrt{a^2 + b^2}\), where 'a' and 'b' are the real and imaginary parts, respectively. Plugging in our values yields \(r = \sqrt{5^2 + 2^2} = \sqrt{29}\). This numeric magnitude is more than a simple calculation; it represents the absolute distance from the origin (0,0) to the point (5, 2) on the complex plane. Understanding magnitude is fundamental in the context of complex analysis and serves as a key component in various operations involving complex numbers.

Argument of Complex Numbers

While the magnitude tells us how far a complex number is from the origin, the argument (or angle) tells us the direction needed to get from the origin to that number. Conceptually, it's the angle a line connecting the origin to the complex number would make relative to the positive real axis.

In our running example, the number 5 + 2i has an argument calculated as \(\theta = arctan(\frac{b}{a})\), where 'a' is the real part and 'b' is the imaginary part of the complex number. Substituting the given values, \(\theta = arctan(\frac{2}{5})\). Although this gives the argument in radians, we often express it in degrees for simplicity. It's important to remember that the argument can have multiple representations, as angles have a periodic nature. This argument aids in expressing complex numbers in their trigonometric form — a succinct representation leveraging the circular nature of trigonometric functions.