Complex numbers can be represented in polar form, which is a method that expresses a complex number in terms of its distance from the origin, denoted as r, and the angle it makes with the real axis, referred to as θ. This representation is particularly useful for visualizing complex numbers on a graph.

To convert a complex number from its standard form x + yi to polar form, we utilize the following equations:

1. The distance r is calculated using the Pythagorean theorem:

\( r = \sqrt{x^2 + y^2} \)

2. The angle θ can be determined using the tangent function:

\( \tan(θ) = \frac{y}{x} \)

To find θ, we use the inverse tangent:

\( θ = \tan^{-1}\left(\frac{y}{x}\right) \)

For example, to convert the complex number 4 + 3i into polar form, we first identify x = 4 and y = 3. We calculate r as follows:

\( r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \)

Next, we find θ:

\( θ = \tan^{-1}\left(\frac{3}{4}\right) \approx 37^\circ \)

Now that we have both r and θ, we can express the complex number in polar form:

\( 4 + 3i = 5(\cos(37^\circ) + i\sin(37^\circ)) \)

It is important to note that the angle θ must be adjusted based on the quadrant in which the complex number lies. For instance:

• If the complex number is in Quadrant I, use θ as calculated.
• If in Quadrant II, add 180 degrees to θ.
• If in Quadrant III, also add 180 degrees.
• If in Quadrant IV, add 360 degrees.

This adjustment ensures that the angle reflects the total rotation from the positive real axis to the point represented by the complex number. Understanding these concepts allows for effective conversion between standard and polar forms of complex numbers, enhancing your ability to work with them in various mathematical contexts.