Problem 34 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 4 $$

Step-by-Step Solution

Answer

The polar form is \( \sqrt{2929}(\cos 5.195 + i\sin 5.195) \).

Step 1: Identify Real and Imaginary Parts

The given complex number is expressed as a difference: \(25 - 48i\). Here, the real part is \(25\) and the imaginary part is \(-48\). Thus, the complex number can be written in rectangular form as \(25 - 48i\).

Step 2: Calculate the Magnitude

The magnitude of a complex number \(a + bi\) is given by \( r = \sqrt{a^2 + b^2} \). So, the magnitude of \(25 - 48i\) is: \[ r = \sqrt{25^2 + (-48)^2} = \sqrt{625 + 2304} = \sqrt{2929}. \]

Step 3: Determine the Argument

The argument \( \theta \) of a complex number \( a + bi \) is given by \( \theta = \tan^{-1}(\frac{b}{a}) \). Since the imaginary part is negative, the number lies in the fourth quadrant, and we use: \[ \theta = \tan^{-1}\left(\frac{-48}{25}\right). \] This will yield an angle close to \(-1.088\) radians. Adding \(2\pi\) gives a positive angle.

Step 4: Convert to Polar Form

Using the magnitude and argument, express the complex number in polar form: \( 4(\cos \theta + i\sin \theta) \). The entire expression simplifies as: \[ \sqrt{2929}(\cos(\theta) + i\sin(\theta)), \] where \(\theta = 2\pi - 1.088 \approx 5.195 \) radians.

Key Concepts

Polar Form of Complex Numbers

Complex numbers have both a real and an imaginary part, like the example given: \(25 - 48i\). While this rectangular form is common, representing complex numbers in polar form can often make calculations easier, particularly for multiplication and division.
The polar form of a complex number expresses it in terms of its magnitude and angle (argument). For a complex number \(a + bi\), its polar form is represented as \(r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the magnitude and \(\theta\) is the argument.

• \(r\) is the distance from the origin to the point in the complex plane.
• \(\theta\) is the angle the line connecting this point to the origin makes with the positive real axis.

Using polar form can simplify many operations like finding roots and products of complex numbers.

Magnitude of Complex Numbers

The magnitude, also known as the modulus, of a complex number gives its distance from the origin in the complex plane. It is calculated using the formula: \( r = \sqrt{a^2 + b^2} \) where \(a\) is the real part and \(b\) is the imaginary part.
For the complex number \(25 - 48i\), the magnitude is calculated as \( \sqrt{25^2 + (-48)^2} = \sqrt{2929} \).
This step is crucial, as the magnitude helps to scale the final polar representation of the complex number.
Understanding the magnitude allows you to effectively map the size of the complex number in the polar form, which directly influences its representation on the complex plane.

Argument of Complex Numbers

The argument of a complex number, represented as \(\theta\), is the angle formed by the vector of the complex number with the positive real axis. For a complex number \(a + bi\), it is calculated using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
In the case of \(25 - 48i\), the argument is determined as \( \tan^{-1}\left(\frac{-48}{25}\right) \), resulting in a negative angle since it lies in the fourth quadrant.
As angles can be represented in multiple ways, to keep the argument within the range [0, 2π], we add \(2\pi\) to this negative angle, yielding \(5.195\) radians.
In essence, the argument helps orient the vector of the complex number within the complex plane, and along with the magnitude, it completely defines the number's position in the polar form.