Problem 33 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 5+5 i $$

Step-by-Step Solution

Answer

Polar form: \(5\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(5 + 5i\). Here, the real part \(a\) is 5, and the imaginary part \(b\) is also 5.

Step 2: Calculate the Magnitude

The magnitude of a complex number \(a + bi\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\). Substituting the values from our complex number: \[r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}.\]

Step 3: Calculate the Argument, \(\theta\)

The argument \(\theta\) is calculated using \( \theta = \tan^{-1}\left(\frac{b}{a}\right)\). For our complex number, this becomes:\[\theta = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1) = \frac{\pi}{4}.\]Since this falls in the first quadrant, no adjustments are needed.

Step 4: Write the Polar Form

The polar form of a complex number is expressed as \(r(\cos \theta + i\sin \theta)\). Substituting the values, we have:\[5\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}).\]

Key Concepts

Polar Form

Complex numbers can be represented in different forms, with the polar form being a particularly insightful one. The polar form expresses a complex number in terms of its magnitude and angle, which provides a geometric understanding of the number. Traditionally, a complex number is denoted as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component. In polar form, the same complex number can be expressed as:

• \( r(\cos \theta + i\sin \theta) \)
• or, using Euler's formula, \( re^{i\theta} \)

Here, \( r \) is the magnitude of the complex number, and \( \theta \) is the argument, or the angle.
This form essentially places the complex number on a plane as a point that can be reached by moving \( r \) units away from the origin in the direction specified by \( \theta \).
It's useful for multiplying and dividing complex numbers, because the operations become simpler when performed in polar form.

Magnitude of a Complex Number

The magnitude, or modulus, of a complex number, provides the distance of the number from the origin in the complex plane. This is similar to finding the length of the hypotenuse in a right triangle. If you have a point \( (a, b) \) on a plane, the magnitude \( r \) is calculated using the Pythagorean theorem:

• \( r = \sqrt{a^2 + b^2} \)

For example, with the complex number \( 5 + 5i \), the magnitude is:

• \( r = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2} \)

This value helps determine how far the number is from the origin, regardless of the direction.
This concept is crucial when converting between rectangular and polar forms.

Argument of a Complex Number

The argument of a complex number indicates its direction on the complex plane. It is the angle, \( \theta \), formed with the positive real axis, usually measured in radians. Finding the argument involves using the inverse tangent function:

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

For the complex number \( 5 + 5i \), the argument is:

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

This calculation shows that the number lies on a line forming a 45-degree angle with the positive x-axis.
When working with arguments, always ensure the angle is adjusted to lie within the specified range, usually from 0 to \( 2\pi \). This step guarantees the right orientation, which is paramount when dealing with complex numbers in various operations.