Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to represent the complex number in standard form. $$9\left(\cos 58^{\circ}+i \sin 58^{\circ}\right)$$

Step-by-Step Solution

Answer

The standard form of the complex number will be sample_result.

Step 1: Understand the Formulas

The complex number in trigonometric form, \( re^{i\Theta} \) can be converted into the standard form, \( a+bi \), using the formula \( a = r*cos(\Theta) \) and \( b = r*sin(\Theta) \). Here, \( r \) represents the modulus of the number and \( \Theta \) represents the angle it makes with the real axis.

Step 2: Convert to standard form

The complex number is given as \( 9 \left(\cos 58^{\circ}+i \sin 58^{\circ}\right) \). Here, \( r=9 \) and \( \Theta = 58^{\circ} \). Substituting the values of \( r \) and \( \Theta \) in the formulas, we get the real part \( a = r \cdot \cos (\Theta) = 9 \cos (58^{\circ}) \) and the imaginary part \( b = r \cdot \sin (\Theta) = 9 \sin (58^{\circ}) \)

Step 3: Use a graphing utility

Next, user can utilize a graphing utility tool to graph out these values. Input the expressions for \( a \) and \( b \) obtained in the last step into the tool and note the output.

Step 4: Write out the final representation

The final standard form will be sample_result as \( a+bi \), where \( a \) and \( b \) are the outputs from the graphing utility.

Key Concepts

Trigonometric Form

The trigonometric form of a complex number is expressed as \( r(\cos \Theta + i \sin \Theta) \). This representation showcases both the magnitude and the direction of the complex number. Here, \( r \) is the modulus, or absolute value, and \( \Theta \), the argument, is the angle the complex number makes with the positive real axis in a complex plane. This form is particularly useful when multiplying or dividing complex numbers, as it simplifies calculations involving angles and magnitudes.

• Modulus \( r \): This is the distance from the origin to the point in the complex plane. It can be calculated as \( \sqrt{a^2 + b^2} \) when the complex number is in standard form \( a + bi \).
• Argument \( \Theta \): The angle can be found using inverse trigonometry functions, typically \( \tan^{-1}(\frac{b}{a}) \).

When given in trigonometric form, it is important to understand how to transition to the standard form to interpret the complex number in terms of real and imaginary components.

Standard Form

Converting a complex number from trigonometric form to standard form \( a+bi \) involves utilizing trigonometric functions of the given angle. To do this:\[ \begin{align*} a & = r \cdot \cos(\Theta) \ b & = r \cdot \sin(\Theta)\end{align*} \]This process entails calculating the real part \( a \) and the imaginary part \( b \) by substituting the modulus and angle into the equations above. Using these formulas, trigonometric expressions like \( 9(\cos 58^{\circ}+i \sin 58^{\circ}) \) can be easily broken down. In this example, the given \( r \) is 9, and the angle \( \Theta \) is 58 degrees.

• Real Part \( a \): Use \( 9 \cos(58^{\circ}) \) to find the projection on the real axis.
• Imaginary Part \( b \): Use \( 9 \sin(58^{\circ}) \) to find the projection on the imaginary axis.

The outcome of these calculations gives you the complex number in a familiar form \( a+bi \), aiding in more straightforward interpretation and graphing.

Graphing Utility

A graphing utility is a powerful tool that allows us to visualize complex numbers and perform various mathematical calculations. After calculating the standard form of a complex number, you can employ a graphing utility to plot it on the complex plane.To graph a complex number:

• Input the calculated real part \( a \) and imaginary part \( b \) into the graphing utility.
• Ensure the graph is set to plot in the complex plane where the x-axis represents the real component and the y-axis represents the imaginary component.

This visualization can help students better understand the magnitude and direction of the complex number from the origin. Graphing utilities can also perform automated transformations like converting forms, thereby validating hand calculations or providing stress-free executions of computationally complex operations.

Real and Imaginary Parts

In the realm of complex numbers, differentiating between the real and imaginary parts is fundamental.- **Real Part (a):** This is the portion of a complex number that aligns with the real-number line in the complex plane. In the expression \( a+bi \), \( a \) is the real part.- **Imaginary Part (b):** This component, when multiplied by \( i \) (the imaginary unit, where \( i^2 = -1 \)), represents the vertical component of the complex number. In \( a+bi \), \( b \) is the imaginary part.When transforming a trigonometric form into the standard form \( a + bi \), you'll extract these parts using:

• \( a = r \cos(\Theta) \)
• \( b = r \sin(\Theta) \)

Understanding and isolating these terms allows for operations like addition, subtraction, and even graphing of complex numbers. Mastery of these parts enhances comprehension of their movement and interaction within the complex plane.