Problem 96 - Algebra and Trigonometry

Question

If you are given a complex number in rectangular form, how do you write it in polar form?

Step-by-Step Solution

Answer

The complex number \(3 + 4i\) in polar form is \(5(cos(53.13°) + i sin(53.13°))\.

Step 1: Identify Rectangular Form

In rectangular form, a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, \(i\) is the imaginary unit, \(a\) is the real part, and \(b\) is the imaginary part. For example, if a complex number in rectangular form is given as \(3+4i\), \(a = 3\) and \(b = 4\).

Step 2: Calculate Magnitude (absolute value/radius) of Complex Number

In polar form, a complex number is written as \(r(cos(\theta) + \isin(\theta))\), where \(r\) is the magnitude, and \(\theta\) is the angle. The magnitude of a complex number is given by the formula \(\sqrt{a^2 + b^2}\). With \(a=3\) and \(b=4\), the magnitude \(r\) is then calculated as \(r=\sqrt{3^2 + 4^2} = 5\).

Step 3: Calculate Angle (argument/phase) of Complex Number

The angle \(\theta\) is calculated using the inverse tangent or, arctan, function: \(\theta = arctan(b/a)\). So in our example, \(\theta = arctan(4/3)\) or \(\theta \approx 53.13°\), depending on whether the result is desired in degrees or radians.

Step 4: Write Polar Form

After finding the magnitude and angle, plug in these values into the polar form: \(r(cos(\theta) + \isin(\theta))\). Using our calculated values, the polar form is \(5(cos(53.13°) + i sin(53.13°))\).