Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$3\left[\cos \left(18^{\circ} 45^{\prime}\right)+i \sin \left(18^{\circ} 45^{\prime}\right)\right]$$

Step-by-Step Solution

Answer

The standard form of the given complex number is \( a + ib \), where \( a \) and \( b \) are obtained in Step 2. The exact values of \( a \) and \( b \) would depend on the precision of calculation.

Step 1: Understanding the given complex number and plotting it

The given complex number is in polar form with magnitude \( r = 3 \) and angle \( θ = 18.75° \). Convert the angle to radians as calculations are more accurate in radians in most programming environments, so, \( θ = 18.75 * (\pi / 180) \) radians. In the complex plane, the real axis is the horizontal axis, and the imaginary axis is the vertical axis. This complex number forms an angle of \( 18.75° \) with the positive direction of the real axis and its distance from the origin is 3. Plot this on the complex plane.

Step 2: Conversion to standard form

The standard form of the complex number is given by \( a + ib \), where \( a = r\cos(θ) \) and \( b = r\sin(θ) \). Here, \( a = 3\cos(18.75°) \) and \( b = 3\sin(18.75°) \). After calculation, get the real part \( a \) and imaginary part \( b \) of the complex number.

Step 3: Writing the final answer

Combine the real part \( a \) and imaginary part \( b \) obtained in the previous step and write the complex number in standard form.

Key Concepts

Polar to Rectangular Conversion

When working with complex numbers, one common task is to convert them from polar form to rectangular form, also known as standard form. Polar form represents a complex number using a magnitude, also called the modulus, and an angle, usually in degrees or radians. In our example, the polar form is given by:

• Magnitude: \( r = 3 \)
• Angle: \( \theta = 18.75^{\circ} \)

To convert to rectangular form, follow these simple steps:
1. Convert the angle from degrees to radians if needed, as in many calculations, radians offer more precision. In our example, that means multiplying \( 18.75 \) by \( \frac{\pi}{180} \).
2. Calculate the real part \( a = r \cos(\theta) \), which gives you how far along the horizontal axis the complex number lies.
3. Calculate the imaginary part \( b = r \sin(\theta) \), showing the distance along the vertical, or imaginary, axis.
The rectangular form is then expressed as \( a + ib \), combining both parts into a more conventional complex number representation.

Graphical Representation of Complex Numbers

Visualizing complex numbers on a plane helps understand their properties and operations better. Every complex number corresponds to a point on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
In our example, the complex number originally given in polar form \( 3\left[\cos(18.75^{\circ}) + i\sin(18.75^{\circ})\right] \) is plotted as follows:

• The real component, or cosine part, dictates the position along the horizontal axis.
• The imaginary component maps onto the vertical axis.

With the magnitude of 3, the point is a distance of 3 units from the origin. The angle \( 18.75^{\circ} \) defines the direction, measured counterclockwise from the positive real axis. By plotting this point, you can see precisely how the number aligns relative to the axes, giving a clear visual representation of the complex number's position and orientation.

Standard Form of Complex Numbers

The transformation from polar to standard form demystifies the expression of complex numbers. The standard form \( a + ib \) displays clearly the real (\( a \)) and imaginary part (\( b \)), and is commonly used for calculations and interpretations.
In our example, completing the conversion gives:

• Real part: \( a = 3\cos(18.75^{\circ}) \), which results in a specific numeric value based on the cosine calculation.
• Imaginary part: \( b = 3\sin(18.75^{\circ}) \), providing the numeric imaginary component.

Together, these values form the final expression in standard form:
\( a + ib \).
This standard form not only simplifies mathematical manipulations, such as addition and subtraction with other complex numbers, but also provides clarity in understanding the magnitude and direction of the complex number relative to the axes in its graphical representation.