Problem 52 - Contemporary Precalculus

Question

In Exercises \(37-52,\) express the number in polar form. $$\sqrt{5}+\sqrt{11} i$$

Step-by-Step Solution

Answer

Question: Express the complex number \(\sqrt{5}+\sqrt{11} i\) in polar form. Answer: \(4\left(\cos\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right)\right) + i \sin\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{ 5}}\right)\right)\right)\)

Step 1: Find the magnitude (modulus) of the complex number

To find the magnitude, we can use the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the complex number, respectively. For the given complex number \(\sqrt{5}+\sqrt{11} i\), we have \(a = \sqrt{5}\) and \(b = \sqrt{11}\). So, the magnitude is calculated as $$ r = \sqrt{(\sqrt{5})^2+(\sqrt{11})^2} = \sqrt{5+11} = \sqrt{16} = 4 $$

Step 2: Find the argument (angle) of the complex number

To find the argument, we can use the formula \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\), where \(a\) and \(b\) are the real and imaginary parts of the complex number, respectively. For the given complex number \(\sqrt{5}+\sqrt{11} i\), we have \(a = \sqrt{5}\) and \(b = \sqrt{11}\). So, the argument is calculated as $$ \theta = \tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right) $$ Note that this angle is not unique, and it can have infinitely many values depending on the number of times \(2\pi\) is added to or subtracted from the initial angle.

Step 3: Write the complex number in polar form

Now that we have calculated the magnitude and the argument, we can write the complex number in polar form \(r(\cos(\theta)+i\sin(\theta))\). Using the values we calculated in the previous steps, we get $$ 4\left(\cos\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right)\right) + i \sin\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right)\right)\right) $$ So, the complex number \(\sqrt{5}+\sqrt{11} i\) is expressed in polar form as $$ 4\left(\cos\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right)\right) + i \sin\left(\tan^{-1}\left(\frac{\sqrt{11}}{\sqrt{5}}\right)\right)\right) $$

Key Concepts

Polar Form

Complex numbers can be represented in polar form, which is especially useful in certain mathematical operations, such as multiplication and division.
In polar form, a complex number is expressed as \( r(\cos(\theta) + i\sin(\theta)) \), where:

• \( r \) is the modulus, representing the distance of the complex number from the origin in the complex plane.
• \( \theta \) is the argument, which indicates the direction of the number as an angle from the positive real axis.

This form relates closely to trigonometry and allows us to see complex numbers as vectors with magnitude and direction.
Using polar coordinates, it becomes easier to handle the complex plane, especially when transforming complex numbers. It offers a clear geometric visualization that can simplify complex mathematical operations.

Argument of Complex Numbers

The argument of a complex number is a crucial element of its polar form.
The argument, often represented as \( \theta \), refers to the angle the complex number makes with the positive real axis in the complex plane.
It is calculated using the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]where \( a \) is the real part and \( b \) is the imaginary part of the complex number.Here are a few key points about the argument:

• The argument can have multiple valid values because angles differ by multiples of \( 2\pi \).
• Typically, the principal value is chosen, which is between \(-\pi\) and \(\pi\).

Understanding the argument helps in locating the complex number accurately in the complex plane and is essential for converting between rectangular form and polar form.

Modulus of Complex Numbers

The modulus, or magnitude, of a complex number is a measure of its size or length in the complex plane. It tells you how far the number is from the origin.
The modulus is denoted by \( r \) and calculated using:\[r = \sqrt{a^2 + b^2}\]where \( a \) is the real part, and \( b \) is the imaginary part of the complex number.Some interesting aspects of modulus include:

• The modulus is always a non-negative number. It can be zero only if the complex number is \( 0 + 0i \).
• For the complex number \( \sqrt{5} + \sqrt{11}i \), we found the modulus to be 4. This tells us that the number is 4 units away from the origin.

The modulus provides valuable information about the scale and comparison of complex numbers, enabling easier manipulation through polar forms.