Problem 13 - Precalculus

Question

Express each complex number in polar form. $$-4+4 i$$

Step-by-Step Solution

Answer

The polar form of \(-4 + 4i\) is \[4\sqrt{2} \left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\] or \[4\sqrt{2} e^{i \frac{3\pi}{4}}.\]

Step 1: Understand the Problem

We need to express the complex number \(-4 + 4i\) in polar form. Polar form of a complex number is given by \(r(\cos \theta + i \sin \theta)\) or \(re^{i\theta}\). To express it in this form, we need to find the magnitude \(r\) and the angle \(\theta\).

Step 2: Finding the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by the formula \(r = \sqrt{a^2 + b^2}\). In this case, \(a = -4\) and \(b = 4\), so \[r = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}.\]

Step 3: Determining the Angle

The angle \(\theta\) is found using \(\tan \theta = \frac{b}{a}\). Here, \(b = 4\) and \(a = -4\), so \[\tan \theta = \frac{4}{-4} = -1.\] This gives us \(\theta = \frac{3\pi}{4}\) since the complex number is in the second quadrant (where both real part is negative and imaginary part is positive).

Step 4: Write the Polar Form

Now that we have the magnitude \(r = 4\sqrt{2}\) and the angle \(\theta = \frac{3\pi}{4}\), we can write the polar form of the complex number as \[4\sqrt{2}\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\] or \[4\sqrt{2} e^{i \frac{3\pi}{4}}.\]

Key Concepts

Complex Number

A complex number consists of two parts: a real part and an imaginary part. In mathematical terms, it is generally expressed as \(a + bi\), where \(a\) represents the real part, and \(b\) is the coefficient of the imaginary part \(i\) (where \(i = \sqrt{-1}\)). For example, in the complex number \(-4 + 4i\), \(-4\) is the real part, and \(4i\) is the imaginary part.

Complex numbers can be visualized on a complex plane. The horizontal axis typically represents the real component, whereas the vertical axis represents the imaginary component. This allows us to treat complex numbers a bit like vectors, helping us analyze them using graphical and mathematical techniques.

Magnitude

The magnitude of a complex number, also known as its modulus, measures its distance from the origin in the complex plane. It can be thought of as the "length" of the vector representing the complex number. For any complex number \(a + bi\), the magnitude \(r\) is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \] In our example with \(-4 + 4i\), the magnitude is found by calculating:

• \(r = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\)

Finding the magnitude helps us understand how "large" or "intense" the complex number is, which is essential when converting to polar form.

Angle (theta)

The angle \(\theta\), sometimes referred to as the argument of the complex number, represents the direction of the vector associated with the complex number in the plane relative to the positive direction of the real axis. This angle is typically found using trigonometric relationships, specifically the tangent function. The formula needed is:\[ \tan \theta = \frac{b}{a} \] In our case of \(-4 + 4i\), since \(b = 4\) and \(a = -4\), we have:

• \(\tan \theta = \frac{4}{-4} = -1\)

Once identified, the angle can be precisely located considering the quadrant in which the complex number lies. For this example, since the real part is negative and the imaginary part is positive, \(\theta\) is in the second quadrant and equals \(\frac{3\pi}{4}\).

This angle is critical for representing the complex number in polar form.

Trigonometry

Trigonometry plays a crucial role in the transformation of complex numbers into their polar forms. It revolves around the relationships between the angles and sides of triangles, particularly right-angled triangles. In the context of complex numbers, trigonometry helps in finding the appropriate angle \(\theta\) for the polar representation.With \(-4 + 4i\), we observe the right triangle formed with:

• "opposite" the imaginary component (4),
• "adjacent" the real component (-4).

Using the basic trigonometric function relationship \(\tan \theta = \frac{opposite}{adjacent}\), we find \(\theta\).

This is key to placing the complex number correctly on the complex plane and ultimately deriving its polar form. Remembering how trigonometric identities work helps in all these transformations.

Polar Coordinates

A core application of polar coordinates is found when complex numbers are expressed using their magnitude and angle, among other fields like physics and navigation. Polar coordinates describe the position of a point on a plane using the 'distance' from a reference point (origin) and 'direction' (angle from a reference direction).When converting from rectangular coordinates \((a + bi)\) to polar form, we use:

• Magnitude \(r = \sqrt{a^2 + b^2} \)
• Angle \(\theta\)

Thus, the complex number \(-4 + 4i\) yields:\[ r( \cos \theta + i \sin \theta) \]\[ 4\sqrt{2}( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) \]Or in exponential form:\[ r e^{i\theta} \]This represents polar form more elegantly as:\[ 4\sqrt{2} e^{i\frac{3\pi}{4}} \]Polar coordinates not only simplify calculations but provide insightful geometric interpretations of complex numbers.