Problem 30 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$1+\sqrt{3}i$$

Step-by-Step Solution

Answer

The polar form is \(2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))\)."

Step 1: Identify the Real and Imaginary Parts

Given the complex number \(1 + \sqrt{3}i\), identify the real part as 1 and the imaginary part as \(\sqrt{3}\).

Step 2: Calculate the Modulus

The modulus \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). For our number, calculate: \[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

Step 3: Find the Argument

The argument \(\theta\) is determined using \(\tan(\theta) = \frac{b}{a}\). Here, \(\tan(\theta) = \frac{\sqrt{3}}{1}\), so \(\tan(\theta) = \sqrt{3}\). The angle \(\theta\) that satisfies \(\tan(\theta) = \sqrt{3}\) is \(\frac{\pi}{3}\) radians, as it is in the first quadrant.

Step 4: Write the Polar Form

The polar form of a complex number is \(r(\cos(\theta) + i\sin(\theta))\). Substituting the values we found: \[ 2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})) \] This is the polar form of the complex number with \(\theta\) between 0 and \(2\pi\).

Key Concepts

Complex Numbers

Complex numbers are a cornerstone of mathematics, merging real and imaginary numbers into a single entity. They are generally written in the form of \(a + bi\), where \(a\) is the real part, and \(bi\) represents the imaginary part with \(i\) being the imaginary unit where \(i^2 = -1\). For instance, the complex number in question, \(1 + \sqrt{3}i\), has a real part of 1 and an imaginary part of \(\sqrt{3}\).
Understanding complex numbers is crucial as they extend the concept of one-dimensional real numbers to the two-dimensional complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This representation allows for a richer understanding of mathematical concepts and phenomena, particularly in areas such as electrical engineering, quantum physics, and applied mathematics.
Whenever you come across a complex number, it’s helpful to think of it as a point in this complex plane, making operations like addition and multiplication straightforward when considering their geometric interpretations.

Modulus

The modulus of a complex number is a measure of its magnitude, similar to the way absolute value measures the magnitude of real numbers. For a complex number \(a + bi\), the modulus \(r\) is calculated as \(\sqrt{a^2 + b^2}\).
Let's see this in action for the complex number \(1 + \sqrt{3}i\):

• For the real part \(a = 1\) and the imaginary part \(b = \sqrt{3}\), the modulus is computed as \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2\).

The modulus can be thought of as the distance from the origin to the point represented by the complex number in the complex plane. This distance is a crucial component in defining the polar form of a complex number, as it corresponds to the length of the vector from the origin to the point. Thus, knowing the modulus helps in understanding not just the size but also the location of a complex number in the complex plane.

Argument

The argument of a complex number refers to the angle that the number makes with the positive real axis in the complex plane. It is usually denoted by \(\theta\).
For a given complex number \(a + bi\), the argument can be found using the tangent function: \(\tan(\theta) = \frac{b}{a}\). This angle \(\theta\) is usually expressed in radians.
In the case of the complex number \(1 + \sqrt{3}i\), we calculate:

• \(\tan(\theta) = \frac{\sqrt{3}}{1} = \sqrt{3}\). This is reminiscent of angles commonly known from trigonometry.
• Here, \(\theta\) corresponds to \(\frac{\pi}{3}\), which is a notable angle in the first quadrant where both sine and cosine values are positive.

Understanding the argument is key for transitioning from Cartesian coordinates to polar coordinates, where angles and magnitudes are used instead of horizontal and vertical offsets. This conversion is particularly useful in complex calculations and visualizations.

Trigonometry

Trigonometry plays a pivotal role in understanding complex numbers, especially when transforming them into their polar form. In polar form, a complex number is expressed as \(r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the modulus and \(\theta\) is the argument.
This form leverages trigonometric identities to clearly describe a complex number's position and magnitude in the complex plane. It’s akin to writing the number in terms of its "size" and "direction" instead of its real and imaginary parts.
For \(1 + \sqrt{3}i\), we have already established:

• \(r = 2\)
• \(\theta = \frac{\pi}{3}\) which correspond to our trigonometric values \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\).

The polar form gives us \(2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))\).
This not only showcases the symmetry and periodicity inherent in trigonometry but also aids in performing complex multiplication and division more intuitively by adjusting modulus and arguments. Thus, trigonometry simplifies complex mathematical operations through graceful and efficient formulations.