Problem 46 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$2(1-i)$$

Step-by-Step Solution

Answer

The polar form is \(2\sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4}))\).

Step 1: Expand the Complex Number

Start by expanding the expression \(2(1-i)\). Distribute the 2, resulting in the complex number \(2 - 2i\).

Step 2: Identify Real and Imaginary Parts

In the complex number \(2 - 2i\), identify and label the real part as \(a = 2\) and the imaginary part as \(b = -2\).

Step 3: Calculate the Magnitude

Use the formula \(r = \sqrt{a^2 + b^2}\) to find the magnitude. Plug in the values: \(r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\).

Step 4: Calculate the Argument

The argument \(\theta\) can be found using \(\tan(\theta) = \frac{b}{a}\). Therefore, \(\tan(\theta) = \frac{-2}{2} = -1\). The angle \(\theta\) whose tangent is -1 and which is in the fourth quadrant is \(\theta = \frac{7\pi}{4}\).

Step 5: Write in Polar Form

Combine the magnitude and the argument to express the complex number in polar form: \(2\sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4}))\).

Key Concepts

Polar Form

The polar form of a complex number provides a unique way of expressing the number based on its magnitude and angle. It is expressed as \[ r(\cos(\theta) + i\sin(\theta)) \] where \( r \) represents the magnitude and \( \theta \) the angle or argument of the complex number. The conversion from the rectangular form, which is \( a + bi \), to the polar form helps in simplifying multiplicative and divisive operations on complex numbers.

• Magnitude \( r \) scales the complex number's size.
• Argument \( \theta \) indicates the direction from the positive real axis.

The polar form is usually preferred when performing operations like power and roots, where it becomes easier to handle the trigonometric functions.

Magnitude

The magnitude of a complex number, often denoted as \( |z| \), represents the distance from the origin to the point \((a, b)\) in the complex plane. To calculate the magnitude, we use the formula\[ r = \sqrt{a^2 + b^2} \]where \( a \) is the real part and \( b \) is the imaginary part of the complex number. For the complex number \( 2 - 2i \), this would be \( \sqrt{4 + 4} = 2\sqrt{2} \).

• Magnitude is always a non-negative real number.
• It signifies the geometric length of the vector representing the complex number.

Remember that this is akin to finding the hypotenuse of a right triangle formed by the real and imaginary components.

Argument of Complex Numbers

The argument of a complex number, denoted \( \theta \), is the angle formed between the positive real axis and the line representing the complex number in the complex plane. It's an essential component when expressing numbers in a polar or trigonometric form. To find the argument, use the formula \[ \tan(\theta) = \frac{b}{a} \]Here, you must consider the quadrant in which your complex number lies to determine the correct angle. For example, with our complex number \( 2 - 2i \), we have \( \tan(\theta) = \frac{-2}{2} = -1 \), placing \( \theta = \frac{7\pi}{4} \) in the fourth quadrant.

• The argument is measured in radians and typically falls between 0 and \( 2\pi \).
• Adjusted for direction, it gives precise information regarding the complex number's orientation.

Trigonometric Form of Complex Numbers

The trigonometric form, synonymous with the polar form, expresses a complex number using trigonometric functions and makes calculations involving powers and roots simpler. Given by \[ r(\cos(\theta) + i\sin(\theta)) \], this form combines both the magnitude and argument of the complex number.

• \( \cos(\theta) \) corresponds to the real part projection on the real axis.
• \( i\sin(\theta) \) represents the imaginary part projection on the imaginary axis.

This formulation is particularly useful when dealing with Euler's formula, where \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). Therefore, the expression can be compactly written as \[ z = re^{i\theta} \], which simplifies many computations involving complex numbers.