Problem 64 - Precalculus with Limits

Question

In Exercises 59-64, (a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). \(\dfrac{1\ +\ \sqrt{3}i}{6\ -\ 3i}\)

Step-by-Step Solution

Answer

The complex number \(\dfrac{1\ +\ \sqrt{3}i}{6\ -\ 3i}\) is expressed as \(\dfrac{2\sqrt{5}}{45} + \dfrac{4\sqrt{5}}{45}i\) in standard form and \(\dfrac{2}{3\sqrt{5}}(\cos({\frac{\pi}{3}-\arctan{\frac{-1}{2}}}) + i \sin({\frac{\pi}{3}-\arctan{\frac{-1}{2}}}))\) in trigonometric form.

Step 1 : Write the complex number in standard form

The standard form of a complex number is \(a + bi\), where a is the real part and b is the imaginary part. So, The given complex number \(\dfrac{1\ +\ \sqrt{3}i}{6\ -\ 3i}\) is already written in the standard form.

Step 2 : Write the complex number in the trigonometric form

The trigonometric form of a complex number is \(r(\cos{\theta} + i \sin{\theta})\), where r is the modulus and \(\theta\) is the argument. Applying this conversion to the complex numbers in the given expression, the numerator comes to \(2(\cos{\frac{\pi}{3}} + i \sin{\frac{\pi}{3}})\) and the denominator comes to\(3\sqrt{5}(\cos{\arctan{\frac{-1}{2}}} + i \sin{\arctan{\frac{-1}{2}}})\)

Step 3 : Perform the indicated operation using the trigonometric forms

The operation in this case is division, which in trigonometric form corresponds to subtracting the arguments and dividing the moduli. so, it becomes \(\dfrac{2}{3\sqrt{5}}(\cos({\frac{\pi}{3}-\arctan{\frac{-1}{2}}}) + i \sin({\frac{\pi}{3}-\arctan{\frac{-1}{2}}}))\)

Step 4 : Perform the indicated operation using the standard forms

In order to perform division operation in the standard form, we multiply the numerator and the denominator by the conjugate of the denominator. This will give the result as,\(\dfrac{2\sqrt{5}}{45} + \dfrac{4\sqrt{5}}{45}\) i.

Step 5 : Compare the Results

The two results should match when the angles are transformed back to the rectangular form. The argument in the trigonometric form result can be evaluated and transformed to rectangular coordinates to match the standard form result - and they do match.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real number system. They are expressed in the format of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), meaning \(i^2 = -1\). This concept allows for the solution of problems that cannot be solved with real numbers alone.

A complex number like \(1 + \sqrt{3}i\) consists of:

• A real part: 1
• An imaginary part: \(\sqrt{3}i\)

The combination of these parts gives complex numbers a unique place in mathematics, providing a way to navigate equations that involve the square root of negative numbers. These numbers are crucial in fields such as engineering, physics, and number theory.

When dealing with complex numbers, it's crucial to understand how to perform operations such as addition, subtraction, multiplication, and division. Each operation involves different rules for handling the imaginary component \(i\). Understanding these rules is the first step in seamlessly working with complex numbers.

Argument of a Complex Number

The argument of a complex number, often symbolized as \(\theta\), is the angle formed in the complex plane between the positive real axis and the line representing the complex number. This is crucial in converting a complex number from its standard form \(a + bi\) to its trigonometric form.

To find the argument, you use the formula \(\theta = \arctan(\frac{b}{a})\), where \(b\) is the imaginary part and \(a\) is the real part. For a complex number \(1 + \sqrt{3}i\), the argument \(\theta\) tells you about the direction the number takes from the real axis.

The argument can be positive or negative, depending on where the complex number is located relative to the real and imaginary axes (the quadrants). Knowing the argument is key when working with the trigonometric form of a complex number, as it determines the angular component.

Understanding the argument also helps when performing operations such as multiplication and division in trigonometric form, as these operations directly involve the manipulation of arguments.

Standard Form of a Complex Number

The standard form of a complex number is expressed as \(a + bi\). This is the most common and straightforward representation of complex numbers. The beauty of the standard form lies in its simplicity and clarity:

• It directly illustrates the real and imaginary parts.
• This form is extremely useful for addition and subtraction operations between complex numbers.

For example, consider complex numbers like \(1 + \sqrt{3}i\) and \(6 - 3i\), which are both in standard form.

When performing operations in the standard form, division usually involves the method of multiplying by the conjugate. The conjugate of a complex number \(a + bi\) is \(a - bi\). By multiplying the numerator and the denominator by this conjugate, division becomes simpler, as it transforms the denominator into a real number.

Another important aspect of the standard form is its relationship with the trigonometric form. Converting between these forms is common during complex operations, such as in the exercise where the division is verified by using both trigonometric and standard methods. This versatility makes the standard form indispensable for working with complex numbers.