Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Real and Imaginary Parts

In the given complex number \(\sqrt{2} - \sqrt{2}i\), the real part is \(\sqrt{2}\) and the imaginary part is \(-\sqrt{2}\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Substitute \(a = \sqrt{2}\) and \(b = -\sqrt{2}\) into the formula: \[r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2.\]

Step 3: Determine the Argument \(\theta\)

The argument \(\theta\) is calculated using the formula \(\theta = \tan^{-1}(\frac{b}{a})\). Substituting \(a = \sqrt{2}\) and \(b = -\sqrt{2}\): \[\theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1).\] Since the point is in the fourth quadrant (as the real part is positive and the imaginary part is negative), we calculate: \[\theta = \frac{7\pi}{4}.\]

Step 4: Write the Polar Form

The polar form of a complex number is given by \(r(\cos\theta + i\sin\theta)\). Substituting our values: \[2\left(\cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right)\right).\] This is the polar form of the complex number.

Key Concepts

Polar Form

The polar form of a complex number is an alternative to the standard Cartesian form and makes use of magnitude and direction (argument). In this form, a complex number is expressed using the magnitude, known as the radius \( r \), and an angle \( \theta \), which is the argument. Instead of focusing on real and imaginary components, polar form allows us to easily handle multiplication and division of complex numbers by harnessing the power of trigonometry.

To write a complex number in its polar form, use the formula:

• \( r (\cos \theta + i \sin \theta) \)

Where \( r \) is the magnitude, and \( \theta \) is the argument of the complex number. This form provides a clear geometric interpretation, representing the complex number as a point in the plane, where \( \theta \) specifies the angle and \( r \) the distance from the origin.

Magnitude

Magnitude of a complex number is essentially how far the point representing the complex number is from the origin of the complex plane. This is similar to finding the length of the hypotenuse of a right triangle.

The formula to find the magnitude, \( r \), of a complex number \( a + bi \) is:

• \( r = \sqrt{a^2 + b^2} \)

It reflects the absolute value of the complex number, representing its size. For the complex number \( \sqrt{2} - \sqrt{2}i \), the magnitude is calculated as:

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

In this case, the magnitude tells us that the complex number is 2 units away from the origin.

Argument of Complex Numbers

The argument \( \theta \) of a complex number is the angle that the line (formed by the origin and the point representing the complex number) makes with the positive direction of the real axis. It provides us with the direction in the complex plane.

Typically, this angle is expressed in radians and can be calculated using the inverse tangent:

• \( \theta = \tan^{-1}(\frac{b}{a}) \)

For \( \sqrt{2} - \sqrt{2}i \), we find:

• \( \theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1) \)

Since the complex number lies in the fourth quadrant (real part positive, imaginary part negative), the correct \( \theta \) considering the quadrant is \( \frac{7\pi}{4} \). This means the direction of this complex number is 7\pi/4 radians from the positive real axis.

Real and Imaginary Parts

The real and imaginary parts are fundamental components of any complex number. A complex number is typically written as \( a + bi \), where:

• \( a \) is the real part, which lies along the horizontal axis of the complex plane.
• \( b \) is the imaginary part, lying along the vertical axis.

For the complex number \( \sqrt{2} - \sqrt{2}i \),

• the real part is \( \sqrt{2} \)
• the imaginary part is \(-\sqrt{2} \)

These parts allow us to plot and visualize the complex number as a point in the complex plane. Understanding these components is crucial for operations and transformations involving complex numbers, such as converting to polar form.