Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Convert Cartesian to Polar Form

First, identify the Cartesian coordinates of the complex number. The complex number is given as \(i(2-2i)\). Simplify it: \(i \times (2 - 2i) = 2i - 2i^2\). Since \( i^2 = -1 \), this becomes \(2i + 2 = 2 + 2i\). The real part \(a = 2\) and the imaginary part \(b = 2\).

Step 2: Calculate the Magnitude

The magnitude (or modulus) \(r\) of the complex number is calculated using the formula: \(r = \sqrt{a^2 + b^2}\). Substitute \(a = 2\) and \(b = 2\): \[r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\].

Step 3: Find the Argument

The argument \(\theta\) is calculated using \(\theta = \tan^{-1} \left( \frac{b}{a} \right)\). With \(a = 2\) and \(b = 2\), \[\theta = \tan^{-1} \left( \frac{2}{2} \right) = \tan^{-1} (1)\]. The angle whose tangent is 1 is \(\frac{\pi}{4}\) (in the first quadrant where both \(a\) and \(b\) are positive).

Step 4: Write in Polar Form

Now that we have both the magnitude and the argument, we can write the complex number in polar form: \( r (\cos \theta + i \sin \theta) \). Thus, \[2\sqrt{2} (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\].

Key Concepts

Polar Form

The polar form of a complex number provides an alternative way to represent it, using its magnitude and angle instead of real and imaginary components. In polar form, a complex number is expressed as \( r (\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude of the complex number.
• \( \theta \) (theta) is the argument or angle, typically in radians.

Understanding polar form makes multiplication and division of complex numbers more intuitive, as the magnitudes are multiplied or divided, and the angles are added or subtracted. It also simplifies plotting complex numbers on the complex plane, especially when dealing with rotations or scaling. By converting a complex number from Cartesian to polar form, we gain insight into both its absolute size and its directional angle.

Cartesian Coordinates

Cartesian coordinates are the standard way to represent complex numbers. They describe a complex number as \( a + bi \), where:

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

These coordinates align with the real and imaginary axes on the complex plane, providing a visual representation. Complex numbers in Cartesian form, like \( 2 + 2i \), result in a point two units along the real axis and two units up on the imaginary axis. This method is beneficial for addition and subtraction, as real parts and imaginary parts can be handled independently. However, for multiplication, division, and understanding the overall magnitude and direction, switching to polar form can be more effective.

Magnitude

The magnitude of a complex number is its size or length, regardless of its direction. It's calculated using the formula:\[ r = \sqrt{a^2 + b^2} \]Where \( a \) and \( b \) are the real and imaginary parts, respectively. In the example given, for the complex number \( 2 + 2i \):

• The real part \( a = 2 \)
• The imaginary part \( b = 2 \)

Substituting these values into the formula gives \( r = \sqrt{8} = 2\sqrt{2} \). The magnitude depicts the distance from the origin to the point in the complex plane. It's similar to finding the hypotenuse in a right-angled triangle, using the Pythagorean theorem. The magnitude provides insight into how "large" or far from the origin a complex number is.

Argument

The argument of a complex number, also known as the angle, indicates its direction on the complex plane. It is calculated using the formula:\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]For the complex number \( 2 + 2i \), the calculation is:

• \( b = 2 \)
• \( a = 2 \)

Thus, \( \theta = \tan^{-1}(1) \), which is \( \frac{\pi}{4} \) radians, as it's in the first quadrant (both real and imaginary parts positive). The argument measures the angle counterclockwise from the positive real axis to the line connecting the origin to the point representing the complex number. The range for \( \theta \) is usually between \( 0 \) and \( 2\pi \) radians. Knowing the argument allows us to visualize the direction or orientation of the complex number in the plane.