Problem 43 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(41-44,\) use a graphing utility to write the $$2\left(\cos 155^{\circ}+i \sin 155^{\circ}\right)$$

Step-by-Step Solution

Answer

The complex number in standard form is approximately \(-1.81+1.04i\)

Step 1: Calculate Real Part

Calculate the real part of the complex number which is given by \(r \cos \theta\). Here \(r = 2\) and \(\theta = 155^\circ\). Use a graphing utility or a calculator to find this number.

Step 2: Calculate Imaginary Part

Calculate the imaginary part of the complex number which is given by \(r \sin \theta\). Here \(r = 2\) and \(\theta = 155^\circ\). Use a graphing utility or a calculator to find this number.

Step 3: Write in Standard Form

Combine the real and imaginary parts to write the complex number in standard form \(a+bi\) where \(a\) is the real part and \(b\) is the imaginary part.

Key Concepts

Standard Form

Complex numbers can be written in several forms, but standard form is one of the most commonly used. The standard form of a complex number is expressed as \(a + bi\).

• \(a\) is the real part of the number.
• \(b\) is the imaginary part, with \(i\) representing the imaginary unit where \(i^2 = -1\).

This form is straightforward and makes it easy to identify the real and imaginary components. When a complex number is given in polar or trigonometric form, we often convert it to standard form.
This conversion allows for easier arithmetic operations and visualization. To convert, simply calculate the real and imaginary parts separately, and then combine them.

Real Part

The real part of a complex number in its standard form \(a+bi\) is represented by \(a\).
In the context of this exercise problem, we are given a complex number in trigonometric form: 2\(\left( \cos 155^{\circ} + i \sin 155^{\circ} \right)\).
The real part can be found using the formula \(r \cos \theta\).
For this problem:

• \(r = 2\),
• \(\theta = 155^{\circ}\).

Now, you can either use a calculator or a graphing utility to compute \(2 \cos 155^{\circ}\).
This calculation gives you the real part of the complex number in the standard form.

Imaginary Part

The imaginary part of a complex number is essential for understanding its full nature. In standard form \(a + bi\), \(b\) represents the imaginary part.
In the given trigonometric form, 2\(\left( \cos 155^{\circ} + i \sin 155^{\circ} \right)\), we calculate this part as \(r \sin \theta\).
Here:

• \(r = 2\),
• \(\theta = 155^{\circ}\).

Using a graphing utility or a calculator, find the value of \(2 \sin 155^{\circ}\).
This value becomes \(b\) in our complex number in standard form. Having both the real and imaginary parts lets us proceed to the final step of conversion.

Graphing Utility

Graphing utilities are powerful tools that can simplify complex number operations. They can graph functions, solve equations, and compute values, making them invaluable in mathematics.
For this problem, a graphing utility helps calculate the values for \(\cos 155^{\circ}\) and \(\sin 155^{\circ}\).
This makes it easy to determine the real and imaginary parts of the complex number. Additionally, graphing utilities can also plot complex numbers on the complex plane. This provides a visual representation, helping students see the relationship between their components. While manual calculations are crucial for understanding, these utilities validate our results and save time.