Complex Number to Rectangular Form Calculator

A complex number in polar form is expressed through its magnitude (r) and phase angle (θ). To translate this into rectangular form, which consists of a real component (a) and an imaginary component (b), we apply Euler's formula and basic trigonometry. The real part represents the horizontal coordinate and is calculated using the formula: a = r × cos(θ). The imaginary part represents the vertical coordinate and is calculated using: b = r × sin(θ). The resulting rectangular notation is written as a + bi.

Imagine you have a complex number with a magnitude (r) of 10 and a phase angle (θ) of 60 degrees (which is π/3 radians). First, ensure your angle is in radians if required, or compute the cosine and sine of 60 degrees directly. The cosine of 60 degrees is 0.5, and the sine is approximately 0.866. Next, calculate the real part: a = 10 × 0.5 = 5. Then, calculate the imaginary part: b = 10 × 0.866 = 8.66. As a result, the polar complex number converts into the rectangular form 5 + 8.66i.

When performing manual conversions or inputting values into your calculator, always double-check whether your angle is set to degrees or radians to avoid drastic calculation errors. Additionally, remember that if your phase angle lies in the second, third, or fourth quadrant, the trigonometric functions will naturally assign the correct negative or positive signs to your real and imaginary coefficients. Finally, retain sufficient decimal precision during intermediate steps to maintain overall accuracy in engineering and physics applications.