Complex Number to Rectangular Form Calculator
Enter a complex number in polar form — magnitude r and angle θ in degrees — to instantly convert it to rectangular form a + bi, where a = r·cos θ and b = r·sin θ.
°
Computed as r · cos(θ)
- 1
Convert angle to radians
53.1301 × π ÷ 180 = 0.927295 - 2
Imaginary part b = r · sin(θ)
5 × sin(53.1301°) = 4 - 3
Real part a = r · cos(θ)
5 × cos(53.1301°) = 3
How does this calculator work?
To convert r∠θ to rectangular form: compute a = r·cos(θ) and b = r·sin(θ), giving the complex number a + bi. Enter the magnitude and angle in degrees and the calculator returns both parts plus the full rectangular expression.
Formula
How this is calculated
A complex number in polar form is written as r∠θ (or r·e^(iθ)), where r is the distance from the origin and θ is the angle from the positive real axis. Euler's formula e^(iθ) = cos θ + i·sin θ connects the two representations: the real part a equals r·cos θ and the imaginary part b equals r·sin θ.
This conversion is the inverse of the rectangular-to-polar transform. The angle is expected in degrees and converted to radians internally before evaluating the trigonometric functions. If the angle is a standard one (0°, 30°, 45°, 90°, 180° etc.) the result is exact by definition, but floating-point evaluation may introduce tiny rounding errors near multiples of π.
Polar-to-rectangular conversion appears constantly in AC circuit analysis (phasors), signal processing (Fourier components), and physics (waves), where computing a + bi is needed for addition or subtraction of complex quantities.
Frequently asked questions
Yes. Angles outside [0°, 360°) are valid — cosine and sine are periodic, so 450° gives the same result as 90°, and −45° gives the same as 315°. The calculator handles any value.
If r = 0 then both a and b are 0, giving the complex number 0 + 0i regardless of the angle. The magnitude must be non-negative; a negative magnitude is not meaningful in standard polar form.
Multiply your radian value by 180/π to convert to degrees first, then enter that. For example, π/4 radians ≈ 45°.
TG we-Calculate Editorial Team. (2026). Complex Number to Rectangular Form Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/complex-number-to-rectangular-form-calculator
TG we-Calculate Editorial Team. "Complex Number to Rectangular Form Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/complex-number-to-rectangular-form-calculator.
TG we-Calculate Editorial Team, "Complex Number to Rectangular Form Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/complex-number-to-rectangular-form-calculator
@misc{wecalculate_complex_number_to_rectangular_form_calculator, title = {Complex Number to Rectangular Form Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/complex-number-to-rectangular-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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