Problem 25 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 1+i $$

Step-by-Step Solution

Answer

The polar form of the complex number is \(\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))\).

Step 1: Identify the Real and Imaginary Parts

The complex number is given as \(1+i\). This means the real part \(a\) is 1, and the imaginary part \(b\) is also 1.

Step 2: Calculate the Modulus

The modulus \(r\) of a complex number \(a + bi\) is given by the formula \(r = \sqrt{a^2 + b^2}\). For \(1 + i\), use \(a = 1\) and \(b = 1\): \[ r = \sqrt{1^2 + 1^2} = \sqrt{2} \].

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

The argument \(\theta\) is the angle formed with the positive direction of the real axis. Using the formula \(\theta = \tan^{-1}(\frac{b}{a})\), find \(\theta = \tan^{-1}(1)\). The angle whose tangent is 1 is \(\frac{\pi}{4}\).

Step 4: Express in Polar Form

A complex number in polar form is expressed as \(r(\cos(\theta) + i\sin(\theta))\). Substitute \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\) into this form: \[\sqrt{2} \left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right)\].

Key Concepts

Complex Number Modulus

Every complex number can be represented in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. To convert this into polar form, we first need to find the modulus of the complex number. The modulus, often denoted by \(r\), is the distance from the origin to the point \((a, b)\) on the complex plane. This is given by the formula:

• \(r = \sqrt{a^2 + b^2}\)

This formula is reminiscent of the Pythagorean theorem. It calculates the length of the hypotenuse in a right triangle formed by the complex number. For example, for the complex number \(1+i\), you plug \(a = 1\) and \(b = 1\) into the modulus formula:

• \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\)

So, the modulus \(r\) is \(\sqrt{2}\), giving you the size or magnitude of the complex number from the origin.

Complex Number Argument

Besides the modulus, we also need to determine the argument when converting a complex number to polar form. The argument, represented by \(\theta\), is the angle formed with the positive x-axis (real line). This angle helps in positioning the complex number on the Argand plane. To compute \(\theta\), use the formula:

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

The argument is basically the angle whose tangent is the ratio of the imaginary part to the real part of the complex number. In our case:

• \(\theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1)\)
• \(\theta = \frac{\pi}{4}\)

Here, \(\theta = \frac{\pi}{4}\) radians, which tells us the direction in which the complex number \(1+i\) lies from the origin. Always ensure \(\theta\) is between \(0\) and \(2\pi\) for polar form.

Trigonometric Form of Complex Numbers

Once both the modulus and argument are known, the complex number can be transformed into its polar or trigonometric form. This expression combines the size and direction into a single, more insightful representation:

• \(r(\cos(\theta) + i\sin(\theta))\)

Here, \(r\) is the modulus and \(\theta\) is the argument. For our specific example of \(1+i\), the values \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\), give us the trigonometric form:

• \(\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))\)

This form is especially useful in calculating powers and roots of complex numbers, and in visualizing them in a geometrical sense. It provides a compact way to handle complex operations, making the polar form indispensable in complex analysis and applications.