Problem 38 - Algebra and Trigonometry

Question

\(25-48=\) 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 \(2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\).

Step 1: Understand the Complex Number

We have the complex number \(\sqrt{3} + i\). Here, \(\sqrt{3}\) is the real part and \(i\) (or \(1\cdot i\)) is the imaginary part.

Step 2: Calculate the Magnitude

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

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

The argument \(\theta\) is found using \(\tan\theta = \frac{b}{a}\). For \(\sqrt{3} + i\), we have:\[ \tan\theta = \frac{1}{\sqrt{3}} \]\(\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\) is a standard angle, resulting in \(\theta = \frac{\pi}{6}\), since \(\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}\).

Step 4: Write the Complex Number in Polar Form

The polar form of a complex number is \(r(\cos\theta + i\sin\theta)\). Substitute \(r = 2\) and \(\theta = \frac{\pi}{6}\):\[ 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right). \]

Key Concepts

Polar Form

Complex numbers have two common representations:

• Rectangular form, such as \(a + bi\)
• Polar form.

The polar form is useful for many mathematical operations, including

• multiplication and division of complex numbers, and
• finding powers and roots.

To convert a complex number to its polar form, you need two things: the magnitude and the argument. The general formula for the polar form of a complex number is \(r(\cos\theta + i\sin\theta)\). Here:

• \(r\) is the magnitude of the complex number, and
• \(\theta\) is the argument of the complex number.

In the given problem, we used the complex number \(\sqrt{3} + i\). We identified the magnitude \(r = 2\) and the argument \(\theta = \frac{\pi}{6}\), finally expressing it as \(2(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})\). This is how you write the polar form of a complex number.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as the modulus, gives us an idea of the size or length of the number in the complex plane. It's similar to finding the length of a vector. For a complex number \(a + bi\), the magnitude \(r\) is determined by the formula \(r = \sqrt{a^2 + b^2}\).

This formula comes from applying the Pythagorean theorem to the rectangular representation of a complex number. In simple terms, if you picture the complex number \(a + bi\) as a point in the complex plane, the magnitude is the distance of this point from the origin.

In our example, with the complex number \(\sqrt{3} + i\), the magnitude is calculated as follows:

• Square the real part \((\sqrt{3})^2 = 3\)
• Square the imaginary part \(1^2 = 1\)
• Add these results: \(3 + 1 = 4\)
• Take the square root: \(\sqrt{4} = 2\)

So, the magnitude of \(\sqrt{3} + i\) is \(2\). This gives us a measure of how far the point is from the origin.

Argument of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. This angle, usually denoted as \(\theta\), helps us describe the direction of the vector that represents the complex number \(a + bi\).

To find this angle \(\theta\), we use the tangent function: \(\tan\theta = \frac{b}{a}\). The argument can be calculated using the inverse tangent function or \(\tan^{-1}(\ldots)\).

In the provided exercise, for the complex number \(\sqrt{3} + i\):

• Real part \(a = \sqrt{3}\)
• Imaginary part \(b = 1\)
• \(\tan\theta = \frac{1}{\sqrt{3}}\)

The exact value of the angle is \(\theta = \tan^{-1}(\frac{1}{\sqrt{3}})\). This gives us a standard angle solution of \(\theta = \frac{\pi}{6}\), since it's a well-known angle for which the tangent value is \(\frac{1}{\sqrt{3}}\).

Understanding how to find and interpret the argument is vital for locating the complex number accurately in the complex plane, and it becomes even more important when dealing with the exponential form of complex numbers.