Problem 40 - 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. $$9.75\left[\cos \left(280^{\circ} 30^{\prime}\right)+i \sin \left(280^{\circ} 30^{\prime}\right)\right]$$

Step-by-Step Solution

Answer

The rectangular form for the given complex number is obtained by calculating the cosine and sine components from the polar form. The complex number is then represented graphically in a plane where the real part is the x-coordinate and the imaginary part is the y-coordinate.

Step 1: Convert the degrees to radians

Before applying the formula, the angle Θ must be converted into radians. The conversion formula is: Rad = \( Θ \times \(\frac{π}{180}\) \). So, Θ = \(280.5 \times \(\frac{π}{180}\) \) rad.

Step 2: Replace Θ and \(r\) in the rectangular form formula

Now we replace \(r\) and Θ in the formula \(r \cos Θ + r i \sin Θ\). Thus, our rectangular form becomes \(9.75 \cos\(\frac{280.5π}{180}\) + 9.75i \sin\(\frac{280.5π}{180}\)\).

Step 3: Find the rectangular form

Next, we compute the cosine and sine components to get the rectangular form representation of the complex number. The cosine of \(\frac{280.5π}{180}\) rad and sine of \(\frac{280.5π}{180}\) rad need to be found.

Step 4: Represent the complex number graphically

To represent the complex number graphically we plot it in the complex plane. The real part represents the x-axis while the imaginary part represents the y-axis. So, the complex number is represented by a point in a plane.

Key Concepts

Standard Form

The standard form of a complex number is one of the most common ways to express complex numbers. In standard form, a complex number is written as

• \( a + bi \)

where \( a \) is the real part and \( bi \) is the imaginary part. Here, \( i \) is the imaginary unit, defined as \( i^2 = -1 \). The formula used in the solution converts polar coordinates to this standard form.
To find the standard form, you must calculate both the cosine and sine components for the given angle. This transforms the polar form \( r[\cos(\Theta) + i \sin(\Theta)] \) into the rectangular format of shelfed standard form \( a + bi \).
For example, starting with

• \( r = 9.75 \)
• \( \Theta = \frac{280.5\pi}{180} \) radians

We compute \( 9.75 \cos(\frac{280.5\pi}{180}) \) for the real part, and \( 9.75i \sin(\frac{280.5\pi}{180}) \) for the imaginary part.

Graphical Representation

Graphical representation is an intuitive method to visualize complex numbers using the complex plane, akin to Cartesian coordinates.
The complex plane treats each complex number \( a + bi \) as a unique point. The horizontal position corresponds to the real part \( a \), whereas the vertical position reflects the imaginary part \( b \).
To graphically represent the complex number from our exercise:

• Calculate the real part obtained from \( 9.75 \cos(\frac{280.5\pi}{180}) \), plot this value along the x-axis.
• Obtain the imaginary component from \( 9.75 \sin(\frac{280.5\pi}{180}) \), position this on the y-axis.

The intersection of these values marks the exact placement of the complex number on the plane, aiding in easy visualization and understanding.

Radians to Degrees Conversion

Converting angles is a crucial step in simplifying problems involving trigonometry and complex numbers.
Degrees and radians are both units used to measure angles. Conversion between them uses the formula:

• \( \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \)

This formula is pivotal when working with trigonometric functions because in calculus, angle measurements are often given in radians.
In the original solution, the angle \( 280.5^\circ \) is converted to radians for use in the trigonometric functions, such as cosine and sine, adhering to this formula. This ensures calculations stay consistent with radian expectations in mathematical analysis.
Knowing this conversion equips you with flexibility to switch between radians and degrees, optimizing understanding across varied mathematical contexts.