Problem 37 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The polar form of the complex number is \(5\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

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 5.

Step 2: Calculate the magnitude

The magnitude \(r\) of a complex number \(a + bi\) is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \] Substitute \(a = 5\) and \(b = 5\) into the formula:\[ r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

Step 3: Determine the angle (argument)

The argument \(\theta\) is found using the formula:\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]Substitute \(a = 5\) and \(b = 5\):\[ \theta = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1) \]Since both \(a\) and \(b\) are positive, the complex number is in the first quadrant, and \(\theta = \frac{\pi}{4}\).

Step 4: Write the number in polar form

To express the complex number in polar form, use the formula:\[ r(\cos \theta + i\sin \theta) \]Substituting \(r = 5\sqrt{2}\) and \(\theta = \frac{\pi}{4}\), we have:\[ 5\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \]

Key Concepts

Complex Numbers

Complex numbers are fascinating numbers that extend our understanding beyond real numbers. They are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) represents the imaginary unit, satisfying \( i^2 = -1 \). The term \( a \) signifies the real part, while \( bi \) is the imaginary part. These numbers find their way into numerous scientific fields, thanks to their versatility in solving equations that reals alone cannot.

• The real part affects the position along the horizontal axis when visualized on the complex plane.
• The imaginary part determines the position along the vertical axis.

When dealing with complex numbers, it's beneficial to represent them geometrically, as it aids in operations such as addition, subtraction, multiplication, and division.

Magnitude of Complex Numbers

The magnitude, or modulus, of a complex number \( a + bi \) measures its distance from the origin in the complex plane. Calculated using the formula \( r = \sqrt{a^2 + b^2} \), it resembles the Pythagorean theorem, where the real part \( a \) and imaginary part \( b \) serve as legs of a right triangle. In this exercise,

• The real part \( a = 5 \)


• The imaginary part \( b = 5 \)


• The magnitude is \( r = \sqrt{5^2 + 5^2} = 5\sqrt{2} \)

Calculating the magnitude is essential in transforming complex numbers from rectangular form to polar form, making them easier to multiply and divide.

Argument of Complex Numbers

The argument of a complex number, denoted as \( \theta \), is the angle formed with the positive real axis in the complex plane. It's calculated using the inverse tangent function: \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
In our problem, since both the real part \( a \) and the imaginary part \( b \) are positive, we know the number is located in the first quadrant,

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

When computing the argument, consider the signs of \( a \) and \( b \) to ascertain the correct quadrant, as they influence the angle value by potentially requiring an adjustment by adding \( \pi \) or \( 2\pi \), depending on the quadrant.

Conversion to Polar Coordinates

Converting a complex number to its polar form involves expressing it as \( r(\cos \theta + i\sin \theta) \). This polar representation is especially useful for complex number operations, as it simplifies multiplication and division.

• The magnitude \( r \) provides the distance from the origin,


• while the argument \( \theta \) specifies the direction.

In our example of \( 5 + 5i \), the polar form becomes \( 5\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \). This form streamlines complex operations similar to polar coordinates in trigonometry, providing a comprehensive understanding of the number's position and behavior in the plane.