Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar form is \( \sqrt{2} (\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) \).

Step 1: Identify the real and imaginary parts

The complex number given is \(-1+i\). Here, the real part \(a\) is \(-1\) and the imaginary part \(b\) is \(1\).

Step 2: Calculate the modulus

The modulus \(r\) of a complex number \(a+bi\) is given by:\[ r = \sqrt{a^2 + b^2} \]Substituting the values of \(a\) and \(b\):\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

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

The argument \(\theta\) can be found using the formula:\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]Substitute \(a = -1\) and \(b = 1\):\[ \theta = \tan^{-1} \left( \frac{1}{-1} \right) = \tan^{-1}(-1) \]Since the complex number is in the second quadrant, where tangent is negative, the argument becomes:\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

Step 4: Write in polar form

The polar form of a complex number is given by:\[ r (\cos \theta + i \sin \theta) \]Substitute \(r = \sqrt{2}\) and \(\theta = \frac{3\pi}{4}\):\[ -1 + i = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

Key Concepts

Polar Form

The polar form of a complex number provides a unique way to express the number using its magnitude and angle. Instead of writing a complex number in the form of \(a + bi\), we describe it with a radius and an angle from the positive x-axis. This is specified as \(r (\cos \theta + i \sin \theta)\), where \(r\) is the modulus (or magnitude), and \(\theta\) is the argument (or angle).

This representation is particularly useful in situations where you need to multiply or divide complex numbers, as it simplifies the process.

• The radius \(r\) tells us how far the number is from the origin in the complex plane.
• The angle \(\theta\) tells us the direction of the number from the positive horizontal axis.

For the complex number \(-1 + i\), converting to polar form involves finding \(r\) and \(\theta\) and then substituting them into the polar form expression. We previously calculated \(r\) as \(\sqrt{2}\) and \(\theta\) as \(\frac{3\pi}{4}\), resulting in \(\sqrt{2} (\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\).

The polar form thus provides a complete geometric view of the complex number, allowing for a deeper understanding and simpler calculations in many situations.

Modulus of Complex Numbers

The modulus of a complex number is essentially the "size" or "length" of the number in the complex plane. It is represented by \(r\) in the polar form equation \(r (\cos \theta + i \sin \theta)\). To determine the modulus of a complex number \(a + bi\), we use the formula:

\[ r = \sqrt{a^2 + b^2} \]

This formula is derived from the Pythagorean theorem, reflecting that the modulus corresponds to the hypotenuse of a right triangle formed by the real and imaginary parts of the complex number.

• In our example of the complex number \(-1 + i\), calculate \(r\) by finding \( \sqrt{(-1)^2 + 1^2} = \sqrt{2} \).
• This value, \(\sqrt{2}\), tells us the distance from the origin to the point \(-1 + i\) in the complex plane.

Understanding the modulus helps to see how "far" a complex number is from the center, making it critical in many mathematical and engineering fields.

Argument of Complex Numbers

The argument of a complex number refers to the angle \(\theta\) formed by the line that represents the complex number with respect to the positive direction of the x-axis (real axis). This angle is measured in radians and varies between 0 and \(2\pi\). Calculating the argument involves using the formula:

\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]

Since the argument should consider the actual position of the complex number in the plane, adjustments are sometimes required based on the quadrant where the number lies.

• In our example, with \(-1 + i\), a direct calculation gives us \(\tan^{-1}(-1)\), but knowing the complex number is in the second quadrant helps us adjust the angle to \( \frac{3\pi}{4} \).
• This adjustment is necessary because tangent's value is the same for angles that differ by \(\pi\). Hence, to stay within the standard interval \([0, 2\pi)\), adjustments are made based on which quadrant the complex number falls into.

Understanding the argument allows us to visualize the angle of elevation from the horizontal axis, providing a comprehensive perspective of the complex number's orientation.