Converting complex numbers from polar form to rectangular form is a straightforward process that involves using the values of the radius (r) and the angle (θ). The rectangular form of a complex number is expressed as x + yi, where x represents the real part and y represents the imaginary part.

To convert a complex number given in polar form, you start by distributing the radius r into the cosine and sine functions of the angle. The general formula for this conversion is:

\[z = r(\cos(\theta) + i\sin(\theta))\]

For example, if you have a complex number in polar form such as 5(cos(37°) + i sin(37°)), you would calculate:

\[z = 5 \cdot \cos(37°) + i \cdot 5 \cdot \sin(37°)\]

Using a calculator, you find that 5 \cdot \cos(37°) \approx 4 and 5 \cdot \sin(37°) \approx 3, leading to the rectangular form:

\[z \approx 4 + 3i\]

In this case, the real part x is 4 and the imaginary part y is 3.

When dealing with angles in radians, the process remains the same. For instance, converting 8(cos(π/6) + i sin(π/6)) involves recognizing the values from the unit circle:

\[\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\]

Thus, the conversion becomes:

\[z = 8 \cdot \frac{\sqrt{3}}{2} - i \cdot 8 \cdot \frac{1}{2}\]

Calculating this gives:

\[z = 4\sqrt{3} - 4i\]

Here, the rectangular form is 4√3 - 4i, with x = 4√3 and y = -4.

In summary, converting from polar to rectangular form involves distributing the radius into the cosine and sine functions, evaluating these expressions, and identifying the real and imaginary parts. This method is efficient and can be performed using either a calculator or values from the unit circle.