Problem 39 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(31-40\) , write the standard form of the complex number. Then represent the complex number graphically. $$5\left[\cos \left(198^{\circ} 45^{\prime}\right)+i \sin \left(198^{\circ} 45^{\prime}\right)\right]$$

Step-by-Step Solution

Answer

The complex number in standard form is \(-4.7 - 1.7i\) and it can be represented graphically by the point (-4.7, -1.7) on the complex plane.

Step 1: Convert into Standard Form

The given complex number is in polar form i.e. \(r[\cos(\theta) + i \sin(\theta)]\) where \(r = 5\) and \(\theta = 198^{\circ} 45^{\prime}\). Using Euler's formula, we know that \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). Thus, our complex number in standard form is given by \[z = 5e^{i198.75 ^{\circ}}\]

Step 2: Convert Degrees into Radians

Before proceeding with the conversion from exponential to rectangular (standard) form, the angle which is presently in degrees needs to be converted into radians. Using the conversion factor \(1 degree = \frac{\pi}{180} radians\), we obtain \(\theta = 198.75^{\circ} * \frac{\pi}{180} = 3.47\) radians. Now our complex number is \[z = 5e^{i3.47}\].

Step 3: Apply Euler's Formula

Applying Euler's formula \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\) to this equation, we get \[z = 5[\cos(3.47) + i\sin(3.47)]\].

Step 4: Calculate Cosine and Sine Values

Now, compute the values of \(\cos(3.47)\) and \(\sin(3.47)\) to the necessary precision. We get \(\cos(3.47) = -0.94\) and \(\sin(3.47) = -0.34\). Substituting these, we get \[z = 5[-0.94 - 0.34i]\].

Step 5: Simplify the Expression

After this, simplify the expression to obtain the standard form of the complex number. The standard form of the complex number is \[z = -4.7 - 1.7i\].

Step 6: Graphical Representation

Plot the point \(-4.7, -1.7\) on the complex plane, which represents the complex number graphically. The x-coordinate corresponds to the real part, and the y-coordinate corresponds to the imaginary part of the complex number.

Key Concepts

Standard Form

Complex numbers can be expressed in standard form as \(a + bi\), where:

• \(a\) is the real part
• \(bi\) is the imaginary part, \(b\) being the coefficient of \(i\), the imaginary unit

To convert a complex number from polar form to standard form, we need to use trigonometric identities. In the earlier exercise, we started with a complex number given in polar format as \(5\left[\cos(198.75^\circ) + i\sin(198.75^\circ)\right]\).
Applying Euler's formula, we calculated cosine and sine of the angle, then simplified to find the standard form \(z = -4.7 - 1.7i\). Here, \(-4.7\) is the real part, and \(-1.7i\) is the imaginary part.

Polar Form

The polar form of a complex number expresses it as a combination of its magnitude and angle. It's written as \(r(\cos(\theta) + i\sin(\theta))\), where:

• \(r\) is the modulus, or magnitude, of the complex number
• \(\theta\) is the argument, or angle, measured in either degrees or radians

In our example, the magnitude \(r\) is 5, and \(\theta\) is given as \(198.75^\circ\).
Polar form is particularly useful when multiplying or dividing complex numbers, or when raising them to powers, as it simplifies the calculations of such operations.

Euler's Formula

Euler's formula is a key link between trigonometry and complex numbers. It states that \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). This powerful equation allows us to express the polar form of a complex number in its exponential form, \(re^{i\theta}\).
In the exercise, we leveraged Euler's formula to transition smoothly from polar to standard form by replacing the trigonometric terms with their exponential equivalents. This formula is particularly advantageous when manipulating complex numbers involving exponential and trigonometric calculations.

Graphical Representation

Visualizing complex numbers relies on plotting them on a complex plane, similar to an x-y graph. Here, the x-axis represents the real part, and the y-axis represents the imaginary part.

• The horizontal axis is the real component.
• The vertical axis is the imaginary component.

For \(-4.7 - 1.7i\), you plot the point \((-4.7, -1.7)\) on the plane. This helps in visualizing both the magnitude and direction of the complex number. By doing this, you can easily interpret and compare the positions and magnitudes of complex numbers visually, much like using vectors in a Cartesian plane.