Problem 42 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ \sqrt{3}+i $$

Step-by-Step Solution

Answer

The polar form is \( 2(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6}) \).

Step 1: Identify the Complex Number Components

The complex number given is \( \sqrt{3} + i \). This number can be expressed in the form \( a + bi \) where \( a = \sqrt{3} \) and \( b = 1 \).

Step 2: Calculate the Magnitude

The magnitude \( r \) of a complex number \( a + bi \) is given by the formula \( r = \sqrt{a^2 + b^2} \). In this case, \( r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \).

Step 3: Calculate the Argument \( \theta \)

The argument \( \theta \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \). For this complex number: \( \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \). Since \( \theta \) is already between 0 and 2\( \pi \), no adjustment is needed.

Step 4: Write in Polar Form

The polar form of a complex number is given by \( r(\cos\theta + i\sin\theta) \). Substituting the values we found: \( 2(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6}) \).

Key Concepts

Complex Number Components

A complex number is generally represented as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit defined by \( i^2 = -1 \). Understanding these components is crucial because they form the basis for further calculations like magnitude and argument.
In our example with \( \sqrt{3} + i \), \( \sqrt{3} \) is the real part and \( 1 \) is the imaginary part. These values help us locate the complex number on the complex plane, much like coordinates do for points in geometry. By determining \( a \) and \( b \), we prepare for calculating other properties like magnitude and argument.

Magnitude of Complex Numbers

Magnitude refers to the distance of the complex number from the origin on the complex plane. This is also known as the modulus. You can think of it as finding the hypotenuse of a right triangle where \( a \) and \( b \) are the base and height.
Mathematically, the magnitude is given by the formula \( r = \sqrt{a^2 + b^2} \). Applying the formula to our complex number \( \sqrt{3} + i \), we have:

• \( a = \sqrt{3} \)
• \( b = 1 \)

Hence, \( r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \). This tells us how 'far' the complex number is from the origin and ensures a correct translation into polar form.

Argument of Complex Numbers

The argument of a complex number is the angle formed with the positive x-axis on the complex plane. It is an essential angle since it helps describe the direction of the complex number from the origin.
Given by \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \), the argument for \( \sqrt{3} + i \) turns out to be:

• \( a = \sqrt{3} \)
• \( b = 1 \)
• \( \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \)

The computed \( \theta \) lies within the standard range of 0 to 2\( \pi \), as required. It defines the angle the line, joining the complex number to the origin, makes with the positive x-axis.

Trigonometric Form

The trigonometric form of a complex number provides a way of expressing complex numbers using trigonometric functions. The form is given by \( r(\cos \theta + i\sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the argument.
For the complex number \( \sqrt{3} + i \), we've calculated:

• Magnitude \( r = 2 \)
• Argument \( \theta = \frac{\pi}{6} \)

Substituting these into the trigonometric form gives \( 2(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6}) \). This representation uses both angle and magnitude to give a complete description of the complex number on the complex plane.

Converting Complex Numbers

Converting between rectangular and polar forms of complex numbers is a powerful tool in complex analysis. While the rectangular form provides straightforward addition and subtraction, the polar form simplifies multiplication and division through trigonometry.
To convert from rectangular to polar:

• First, find the magnitude using \( r = \sqrt{a^2 + b^2} \).
• Then, calculate the argument via \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
• Express the number as \( r(\cos \theta + i\sin \theta) \).

In our example, the conversion gives us a neat representation of the complex number utilizing both magnitude and argument, allowing for more effortless operation involving powers and roots of complex numbers.