Complex Number to Rectangular Form Calculator

Rectangular form expresses a complex number as a real part plus an imaginary part.

How the Math Works

Complex numbers can be expressed in polar form as r∠θ, where r is the magnitude (distance from origin) and θ is the angle from the positive real axis. To convert to rectangular form a + bi, we use trigonometry: the real part a equals r multiplied by the cosine of θ, while the imaginary part b equals r multiplied by the sine of θ. This works because cosine gives the horizontal component (adjacent side) and sine gives the vertical component (opposite side) of a right triangle with hypotenuse r and angle θ.

Practical Applications

To use this conversion in practice, first ensure your angle is in radians if your calculator is set to radian mode, or convert degrees to radians by multiplying by π/180. Enter the magnitude r and angle θ into the calculator, which automatically computes a = r·cos θ and b = r·sin θ. For example, converting 5∠30° yields a = 5·cos(30°) = 4.33 and b = 5·sin(30°) = 2.5, giving the rectangular form 4.33 + 2.5i.

Day-to-Day Use

This conversion is essential in electrical engineering for analyzing AC circuits, where impedance and phase angles are more naturally expressed in polar form but calculations often require rectangular form. It's also valuable in physics for representing waveforms, oscillations, and quantum states, and in computer graphics for rotating objects and performing 2D transformations. Students benefit from mastering this conversion as it bridges abstract mathematical concepts with practical problem-solving across STEM fields.

Worked example

2∠60° → 1 + 1.732i.

FAQ