Problem 43 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 3+4 i $$

Step-by-Step Solution

Answer

The polar form of \(3 + 4i\) is \(5e^{i0.927}\).

Step 1: Identify the Components

The given complex number is \( 3 + 4i \). Here, the real part is \( 3 \) and the imaginary part is \( 4 \).

Step 2: Calculate the Magnitude

The magnitude (or modulus) of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). For our complex number, this becomes \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 3: Calculate the Argument

The argument \( \theta \) of a complex number \( a + bi \) is given by \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \). For our number, we have \( \theta = \tan^{-1} \left( \frac{4}{3} \right) \). This calculation results in \( \theta \approx 0.927 \) radians, as \( \tan^{-1}(1.333) \approx 0.927 \) radians, and it lies within \([0, 2\pi)\).

Step 4: Write in Polar Form

A complex number in polar form is expressed as \( r(\cos \theta + i \sin \theta) \) or \( r e^{i\theta} \). Therefore, \( 3 + 4i \) in polar form is \( 5 \left( \cos(0.927) + i \sin(0.927) \right) \) or \( 5e^{i0.927} \).

Key Concepts

Complex Number Magnitude

The magnitude of a complex number, often referred to as the modulus, is a fundamental property that gives us an idea of the 'size' or 'length' of the number when plotted on a complex plane. To find the magnitude of a complex number of the form \(a + bi\), where \(a\) and \(b\) are real numbers, we use the formula:
\[|z| = \sqrt{a^2 + b^2}\]This formula is akin to the Pythagorean theorem and helps in finding the straight-line distance from the origin \((0,0)\) to the point \((a, b)\) in the complex plane.

For example, consider the complex number \(3 + 4i\). To compute its magnitude, we plug the values into the formula:
\[|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]Thus, the magnitude of the complex number \(3 + 4i\) is 5.

• Magnitude helps in comparing sizes and understanding positioning in polar coordinates.
• It remains the same even if the complex number is multiplied by \(-1\).

Complex Number Argument

The argument of a complex number is the angle it makes with the positive real axis on the complex plane. The argument is crucial for converting complex numbers into polar form as it gives direction. For a complex number \(a + bi\), the argument, denoted as \(\theta\), can be calculated using:
\[\theta = \tan^{-1}\left( \frac{b}{a} \right)\]This is based on the tangent function, which relates the angle of a right triangle to its opposite and adjacent sides.

Consider the complex number \(3 + 4i\):

• Here, \(a = 3\) and \(b = 4\).
• The argument is \(\theta = \tan^{-1}\left(\frac{4}{3}\right)\), approximately \(0.927\) radians.

The result must fall within the range \([0, 2\pi)\), which describes a full rotation on the complex plane.

Remember, the quadrant determines the specific procedures:

• First Quadrant: Use the formula directly.
• Second/Third/Fourth Quadrants: Adjust using \(+\pi\) or other rules.

Having the correct argument ensures accurate polar representation and helps in understanding complex number behavior better, particularly when used in trigonometric identities or complex multiplication.

Convert Complex to Polar Form

Converting a complex number to its polar form gives a unique and insightful representation. This form utilizes two key components: the magnitude and the argument. A complex number \(z = a + bi\) is expressed in polar form as:
\[z = r(\cos \theta + i \sin \theta)\]or alternatively with Euler's formula:
\[z = r e^{i\theta}\]

• Here, \(r\) is the magnitude \(|z|\).
• \(\theta\) is the argument found in radians.

Let's convert \(3 + 4i\) into polar form:

• We previously calculated the magnitude \(r = 5\).
• The argument \(\theta \approx 0.927\) radians.
• Thus, the polar form becomes \(5(\cos 0.927 + i \sin 0.927)\) or concisely \(5 e^{i 0.927}\).

The polar form is particularly useful for multiplicative operations and clarifies relationships between complex numbers, often simplifying calculations involving powers and roots.