Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to represent the complex number in standard form. $$2\left(\cos 73^{\circ}+i \sin 73^{\circ}\right)$$

Step-by-Step Solution

Answer

The complex number is represented on the graph as the point with coordinates \((2×\cos(\frac{73×π}{180}), 2×\sin(\frac{73×π}{180}))\)

Step 1: Understand the trigonometric form of the complex number

The trigonometric form of a complex number is given by \( r(\cos θ + i\sin θ) \) where \( r \) is the modulus and \( θ \) is the argument of the complex number. In the given complex number, \( r = 2 \) and \( θ = 73° \)

Step 2: Convert the argument into radians

Since most graphing utilities use radians, we need to convert the argument from degrees to radians. Therefore, \( θ = 73° = \frac{73×π}{180} radians \)

Step 3: Calculate the coordinates

The complex number in trigonometric form can be represented on a graph as a point with coordinates \((r\cos θ, r\sin θ)\). Therefore, we need to calculate these coordinates. Using \( r = 2 \) and \( θ = \frac{73×π}{180} radians \), the coordinates are \((2×\cos(\frac{73×π}{180}), 2×\sin(\frac{73×π}{180}))\)

Step 4: Plot the coordinates

Now plot the point with coordinates \((2×\cos(\frac{73×π}{180}), 2×\sin(\frac{73×π}{180}))\). This point will represent the position of the complex number on the graph.

Key Concepts

Trigonometric Form of Complex Numbers

Complex numbers can be elegantly expressed in trigonometric form, which is especially useful for visualizing and performing calculations involving rotations. This form is given by the expression: \( r(\cos \theta + i\sin \theta) \). The

• \( r \) is known as the modulus, representing the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument, indicating the angle measured counterclockwise from the positive real axis.

For example, in the complex number given, \( r = 2 \) and \( \theta = 73^{\circ} \). This means the complex number represents a point lying a distance of 2 units from the origin at an angle of 73 degrees. Understanding this representation can simplify complex number operations, especially multiplication and division.

Converting Degrees to Radians

Radians are often more convenient and are used by most mathematical tools, including graphing utilities. To convert from degrees to radians, you multiply the number of degrees by \( \frac{\pi}{180} \).

• Remember, \( 180^{\circ} \) is equivalent to \( \pi \) radians.
• This conversion utilizes the circumference of a unit circle.

Given \( 73^{\circ} \), the conversion becomes: \[73^{\circ} = \frac{73 \times \pi}{180} \text{ radians}\]Knowing how to convert between degrees and radians is crucial for using various mathematical tools effectively, including plotting complex numbers.

Understanding Graphing Utilities for Complex Numbers

Graphing utilities are tools that help visually represent mathematical entities, including complex numbers. When you plot a complex number using graphing utilities, you're typically plotting a point on the complex plane where:

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

For the complex number \( 2(\cos \theta + i \sin \theta) \), the point you'll plot has coordinates \((2\cos\theta, 2\sin\theta)\).This means you:

• Calculate \( 2 \cos\theta \) for the x-coordinate.
• Calculate \( 2 \sin\theta \) for the y-coordinate.

Using these calculations, you can easily plot the complex number on the graph, enabling visual analysis and a better understanding of complex relationships.