Problem 45 - Precalculus with Limits

Question

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(2(\cos\ 155^{\circ} + i\ \sin\ 155^{\circ})\)

Step-by-Step Solution

Answer

The complex number in standard form is \(a + bi\), where \(a\) and \(b\) are the values calculated in Step 2.

Step 1: Identify the polar form of complex number

The given complex number is in polar form: \(2(\cos\ 155^{\circ} + i\ \sin\ 155^{\circ})\). This format can be written as \(r(\cos θ + i\sin θ)\), where \(r\) is the magnitude (2 in this case) and \(θ\) represents the angle in degrees (155 degrees in this case).

Step 2: Compute the real and imaginary parts

The real part \(a\) and imaginary part \(b\) of the complex number in standard form \(a + bi\) can be calculated as: \[a = r\cos θ,\] \[b = r\sin θ.\] Here, \(r = 2\) and \(θ = 155^{\circ}\). Using a calculator or graphing utility, compute \(a = 2\cos(155^{\circ})\) and \(b = 2\sin(155^{\circ})\).

Step 3: Express the complex number in standard form

With the values of \(a\) and \(b\) computed in the previous step, write the complex number in the standard form \(a + bi\).

Key Concepts

Polar Form

Understanding the polar form of complex numbers is important in mathematics. Polar form expresses a complex number using two values: the magnitude

• which tells us how "long" the number is in a geometric sense
• and the angle

which tells us the direction. In the exercise, the polar form is given as \(2(\cos 155^\circ + i\sin 155^\circ)\). Here,
\(2\) represents the magnitude or the "distance" from the origin on a polar coordinate system.

• Magnitude \(r\): This is the 2 in our given problem, indicating how far the point is from the origin.
• Angle \(θ\): Our angle is \(155^\circ\), which indicates the clockwise rotation from the positive horizontal axis to reach the complex number's point. This angle shows us exactly where our complex number is located on the plane if measured in degrees.

Switching from polar form to standard form requires converting these polar coordinates to cartesian coordinates which can provide insight into plotting and understanding complex numbers.

Standard Form

When dealing with complex numbers, the standard form refers to the algebraic expression \(a + bi\), where

• \(a\) is the real part
• and \(b\) is the imaginary part.

This form makes it easy to perform arithmetic operations on complex numbers. To convert our complex number from polar to standard form, we utilize the relationships \[a = r\cos(θ),\] \[b = r\sin(θ).\] From the exercise, \(r\) equals 2, and \(θ\) is \(155^\circ\). Using these, we calculate

• the real part \(a = 2\cos(155^\circ)\)
• and the imaginary part \(b = 2\sin(155^\circ)\).

Thus, we write the complex number in its standard form as \(a + bi\). This transformation is essential for carrying out further operations such as addition or multiplication of complex numbers.

Real and Imaginary Parts

In any complex number given in standard form \(a + bi\), the real and imaginary parts each play crucial roles. The real part \(a\) and the imaginary part \(b\) both arise naturally through the combination of polar coordinates. In our example from the exercise, we calculated

• \(a = 2\cos(155^\circ)\)
• and \(b = 2\sin(155^\circ)\).

These values represent a move horizontally and vertically respectively in the complex plane.

• The real part \(a\) can be seen as a simple shift left or right along the x-axis, representing the horizontal component of the number's position.
• The imaginary part \(b\) shifts the point up or down along the y-axis, representing the vertical component from 0 on the imaginary axis.

In practical terms, these parts allow you to understand and visualize complex numbers both in everyday calculations and when plotting them on the complex plane.