Problem 42 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(9.75[\cos(280^{\circ}30') + i\ \sin(280^{\circ}30')]\)

Step-by-Step Solution

Answer

The standard form of the complex number is \(-1.285 - 9.604i\).

Step 1: Converting Polar Form to Standard Form

The standard form of a complex number can be attained from its polar form using the following relations: \(z = r\cos(\theta) + ir\sin(\theta)\). Here, \(z\) is the complex number, \(r\) is the absolute value of \(z\), and \(\theta\) is the argument of \(z\). For the complex number in question, \(r=9.75\) and \(\theta=280^{\circ}30'\). As 1 degree contains 60 minutes, the angle in degrees would be \(280^{\circ} + \frac{30'}{60} = 280.5^{\circ}\). Converting this to radians, we get \(\theta = \pi \times \frac{280.5}{180} =1.648053 rad\). So, the standard form can be achieved by putting the value of \(r\) and \(\theta\) into the above relation.

Step 2: Calculating the Real and Imaginary Parts

Now calculate the real part which is \(r\cos(\theta) = 9.75 \times cos(1.648053) \approx -1.284948\) and the imaginary part which is \(r\sin(\theta) = 9.75 \times sin(1.648053) \approx -9.603577\). So, the standard form of the given complex number is \(Z = -1.285 - 9.604i\).

Step 3: Graphing on the Complex Plane

The graph of the complex number is a point in the complex plane, where the X-axis represents the real part and the Y-axis represents the imaginary part. Mark the point \((-1.285, -9.604)\) and this represents the provided complex number.

Key Concepts

Polar Form

When discussing complex numbers, the polar form is an alternative way of expressing these numbers using a distance and an angle, rather than just the real and imaginary components. This form is expressed as

• \( z = r(\cos\theta + i\sin\theta) \)

where \( r \) is the magnitude (or modulus) of the complex number, representing the distance from the origin to the point in the complex plane. The angle \( \theta \) is the argument of the complex number, measured from the positive real axis. The polar form is particularly useful for multiplication and division of complex numbers, as these operations are more straightforward in polar form than in standard form. Converting from polar form to standard form involves calculating the real and imaginary parts using the equations

• Real part: \( r \cos \theta \)
• Imaginary part: \( r \sin \theta \)

These expressions stem from the basic trigonometric relationships within the unit circle.

Standard Form

Complex numbers are most commonly expressed in their standard form, as

• \( z = a + bi \)

where \( a \) is the real component and \( b \) is the imaginary component of the complex number. Transitioning from polar to standard form involves using trigonometry to find the exact numeric values of these components. Given a complex number in polar form, utilizing the magnitude \( r \) and angle \( \theta \), you compute the real part as \( a = r \cos(\theta) \) and the imaginary part as \( b = r \sin(\theta) \). For example, if \( r = 9.75 \) and \( \theta = 280.5^{\circ} \), the real part \( a \) and the imaginary part \( b \) can be found using the cosine and sine of the angle in radians, respectively. This conversion enables an easy representation on the complex plane.

Graphing Complex Numbers

Graphing complex numbers transforms them from an abstract concept into a visual entity by placing them on a complex plane, which greatly aids in comprehension. The complex plane is composed of a horizontal axis, representing the real part, and a vertical axis, representing the imaginary part. Every point on this plane corresponds to a specific complex number.To graph a complex number \( z = a + bi \):

• Locate \( a \) on the horizontal (real) axis.
• Locate \( b \) on the vertical (imaginary) axis.
• The intersection of these coordinates marks the complex number's position.

In practice, for the complex number \( Z = -1.285 - 9.604i \), you would start from the origin (0,0), move left to \( -1.285 \) along the real axis, then move down to \( -9.604 \) along the imaginary axis. This plotting process not only marks the location but provides insight into the relationship between the components and the whole, showing the distance and direction from the origin.