Problem 44 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ -3-3 i $$

Step-by-Step Solution

Answer

The polar form is \\(3\sqrt{2} \text{cis} \frac{5\pi}{4}\\).

Step 1: Convert to Complex Number Form

The complex number we are working with is \( -3 - 3i \). This means the real part is -3, and the imaginary part is also -3.

Step 2: Calculate Magnitude

The magnitude of the complex number, also called the modulus, is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\) where \(a\) is the real part and \(b\) is the imaginary part. In this case, \(|z| = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\).

Step 3: Determine Argument Angle

The argument \(\theta\) is found using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Here, \(b = -3\) and \(a = -3\), so \(\theta = \tan^{-1}\left(\frac{-3}{-3}\right) = \tan^{-1}(1)\). The principal value of \(\tan^{-1}(1)\) is \(\frac{\pi}{4}\). Considering which quadrant the complex number falls (both parts are negative, so the third quadrant), the angle is \(\pi + \frac{\pi}{4} = \frac{5\pi}{4}\).

Step 4: Express in Polar Form

Combine the magnitude and the argument to express the complex number in polar form: \(3\sqrt{2} \text{cis} \frac{5\pi}{4}\) where \(\text{cis} \theta = \cos \theta + i \sin \theta\).

Key Concepts

Complex Numbers

A complex number is an essential concept in mathematics, particularly when dealing with quantities that cannot be solely expressed in real numbers. Typically, a complex number is written as \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). Complex numbers are represented on the complex plane, which resembles a two-dimensional graph where:

• The horizontal axis represents the real part.
• The vertical axis represents the imaginary part.

For example, the complex number \( -3 - 3i \) has

• a real part of \(-3\), and
• an imaginary part of \(-3\).

This allows us to plot the number at the point \((-3, -3)\) on the complex plane.

Magnitude

The magnitude of a complex number, often referred to as its modulus, measures its distance from the origin of the complex plane. This is somewhat analogous to finding the length of the hypotenuse in a right-angled triangle. If a complex number is given by \( a + bi \), its magnitude is calculated using the formula:\[|z| = \sqrt{a^2 + b^2}\]Where:

• \( a \) is the real part, and
• \( b \) is the imaginary part.

For the complex number \( -3 - 3i \), the magnitude is:\[|z| = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\]This magnitude tells us that the complex number is \( 3\sqrt{2} \) units away from the origin in the complex plane.

Argument Angle

The argument angle of a complex number provides its direction in the complex plane, similar to a clock's hand pointing at a time. To find this angle in polar coordinates, known as \( \theta \), we typically use:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]Where:

• \( b \) is the imaginary part, and
• \( a \) is the real part.

For \( -3 - 3i \), since both parts are negative, indicating the number is in the third quadrant, the calculation gives:

• \( \theta = \tan^{-1}(1) = \frac{\pi}{4} \)
• Adjusted for the third quadrant: \( \pi + \frac{\pi}{4} = \frac{5\pi}{4} \)

This accounts for the full angular span considering both negative components.

Trigonometry

Trigonometry is a significant element when dealing with complex numbers in polar form. It involves using trigonometric functions to express these numbers clearly. The polar form of a complex number \( a + bi \) is expressed as:\[r \text{cis} \theta = r (\cos \theta + i \sin \theta)\]Where:

• \( r \) is the magnitude.
• \( \theta \) is the argument angle.

For \( -3 - 3i \), with a magnitude of \( 3\sqrt{2} \) and an argument of \( \frac{5\pi}{4} \), the polar form becomes:\[3\sqrt{2} \text{ cis } \frac{5\pi}{4} = 3\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})\]This expression aids in visualizing the directional and magnitude properties of the complex number beautifully and compactly.