Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar form is \( 2( \cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})) \).

Step 1: Identify the complex number

The given complex number is \( z = \sqrt{2} + \sqrt{2}i \). This can be written in the form \( a + bi \), where \( a = \sqrt{2} \) and \( b = \sqrt{2} \).

Step 2: Calculate the modulus

The modulus \( r \) of a complex number \( z = a + bi \) is given by \( r = \sqrt{a^2 + b^2} \). Substitute \( a \) and \( b \) to find the modulus: \[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \].

Step 3: Calculate the argument

The argument \( \theta \) of a complex number \( z = a + bi \) can be found using \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \). Here, substitute \( a \) and \( b \): \[ \theta = \tan^{-1} \left( \frac{\sqrt{2}}{\sqrt{2}} \right) = \tan^{-1}(1) = \frac{\pi}{4} \].

Step 4: Write the polar form

The polar form of a complex number is given by \( z = r(\cos(\theta) + i\sin(\theta)) \). Substituting the values found: \[ z = 2 \left( \cos\left( \frac{\pi}{4} \right) + i\sin\left( \frac{\pi}{4} \right) \right) \].

Key Concepts

Polar Form

Complex numbers can be represented in many different ways. One of the most useful forms is the **polar form**. In this representation, a complex number is expressed using its magnitude and angle, instead of its real and imaginary components.The polar form of a complex number is written as:

• \( z = r(\cos(\theta) + i\sin(\theta)) \)

Where:

• \( r \) is the modulus of the complex number, which indicates its distance from the origin
• \( \theta \) is the argument, representing the angle the complex number makes with the positive x-axis

Expressing a complex number in polar form can simplify calculations like multiplication and division.It is particularly helpful in understanding the geometric interpretation of complex numbers.

Modulus

The **modulus** of a complex number is akin to its absolute value. It tells us how "large" or "far" the complex number is from the origin in the complex plane.If you have a complex number, say \( z = a + bi \), the modulus is found using the formula:

• \( r = \sqrt{a^2 + b^2} \)

For instance, for \( z = \sqrt{2} + \sqrt{2}i \), the modulus is:

• \( r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{4} = 2 \)

The modulus provides significant information about the vector magnitude of the complex number, a crucial aspect when dealing with complex multiplication or division.

Argument

The **argument** of a complex number is the angle it forms with the positive x-axis of the complex plane. This angle lets us know the direction of the complex number from the origin.To find the argument, usually denoted by \( \theta \), of a complex number \( z = a + bi \), use the formula:

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

In our example with \( z = \sqrt{2} + \sqrt{2}i \), the argument is:

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

It's crucial to ensure \( \theta \) falls between 0 and 2\( \pi \) to make the angle standardized across calculations.The argument, together with the modulus, forms the basis of the polar representation.

Trigonometry

**Trigonometry** plays a central role in the polar form of complex numbers.This is because the cosine and sine functions are involved in expressing the angle or direction of the complex number in the complex plane.In polar form, the cosine part \( \cos(\theta) \) influences the real part of the complex number, whereas the sine part \( \sin(\theta) \) influences the imaginary part.When we express a complex number as:\( z = r(\cos(\theta) + i\sin(\theta)) \), we're inherently using trigonometry.This usage simplifies many operations like rotations and simplifies the multiplicationor division of complex numbers.Understanding trigonometry helps us make sense of why and how the angle of a complex number shifts, which is vital when manipulating these numbers in various applications.