Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$6\left[\cos \left(230^{\circ} 30^{\prime}\right)+i \sin \left(230^{\circ} 30^{\prime}\right)\right]$$

Step-by-Step Solution

Answer

The standard form of the given complex number is \(-3.06 - 4.46i\). Graphically, this number is represented as a point located at \(-3.06\) on the x-axis and \(-4.46\) on the y-axis of the complex plane.

Step 1: Interpret the given complex number

The given complex number is \(6[\cos (230^{\circ} 30^{\prime})+i \sin(230^{\circ} 30^{\prime})]\). Here, \(r=6\), and the angle \(\theta\) is \(230^{\circ} 30^{\prime}\). Covert this into decimal by dividing the minutes part (30) by 60 to become \(230.5^\circ\). So our \(\theta=230.5^\circ\).

Step 2: Transform into standard form

The standard form is \(a+bi\) where \(a = r\cos \theta\) and \(b = r\sin \theta\). So, \(a = 6 \cdot \cos(230.5^\circ)\) and \(b = 6 \cdot \sin(230.5^\circ)\). Calculate these values to obtain \(a\) and \(b\).

Step 3: Convert degrees to radians

To calculate the values of \(a\) and \(b\), we need to convert degrees to radians as trigonometric functions in computers and calculators use angles in radians. The conversion is \(-3.92\) radians.

Step 4: Calculate a and b

Using the values of \(r\) and \(\theta\), calculate \(a\) and \(b\). Which further simplifies to \(a=-3.06\) and \(b=-4.46\).

Step 5: Present complex number in standard form

Presenting the calculations from the prior step as \(a+bi\) results in the standard form for the given complex number: \(-3.06 - 4.46i\)

Step 6: Graphical representation

Plot the value \(-3.06\) on the x-axis and \(-4.46\) on the y-axis to represent the standard form of the given complex number on a graph.

Key Concepts

Graphical Representation of Complex Numbers

Complex numbers can be visually understood by plotting them on a graph. This involves identifying any complex number as a point in the two-dimensional Cartesian coordinate plane. The real part of the complex number corresponds to the x-coordinate (horizontal axis), while the imaginary part is the y-coordinate (vertical axis). This graphical representation can greatly aid in the understanding and computation of complex numbers.

For example, when you have a complex number in the form of \(a + bi\), plotting it involves:

• Placing the x-coordinate at "a," which represents the real part.
• Placing the y-coordinate at "b," representing the imaginary part.

For the complex number from our example, \(-3.06 - 4.46i\), it is plotted by:

• Marking -3.06 on the x-axis
• Marking -4.46 on the y-axis

This position visually captures the size and direction of the complex number in the plane.

Standard Form of Complex Numbers

The standard form of complex numbers presents these numbers as a combination of a real part and an imaginary part. Written as \(a + bi\), where "a" is the real component and "b" is the imaginary part multiplied by 'i', the imaginary unit. This form makes it straightforward to perform arithmetic operations on complex numbers.

In our given problem, you started with the number expressed in terms of cosine and sine functions: \([6 \cos (230.5^\circ) + i \sin(230.5^\circ)]\). To convert it to the standard form, follow these steps:

• Calculate the real part as \(a = r \cos \theta\)
• Calculate the imaginary part as \(b = r \sin \theta\)
• Here, \(r = 6\) and \(\theta = 230.5^\circ\).

After performing these calculations, you find that \(a = -3.06\) and \(b = -4.46\), giving the standard form \(-3.06 - 4.46i\). This form is particularly useful for further calculations and analysis.

Radians and Degrees Conversion

Radians and degrees are both units for measuring angles. In mathematics, especially in trigonometry, radians are often preferable because they allow for simpler derivatives and integrals among many other advantages.

When working with trigonometric functions in many calculators and programming environments, angles are required in radians rather than degrees. The conversion is simple:

• Knowing that 360 degrees equals \(2\pi\) radians
• The formula is: \( ext{radians} = ext{degrees} \times \frac{\pi}{180}\)

For example, the given angle of \(230.5^\circ\) when converted into radians is approximately \(-3.92\) radians. This conversion is crucial for accurately performing trigonometric calculations in many mathematical and computational settings.