Polar to Complex Calculator

This polar to complex calculator converts a complex number from polar form into its rectangular form, giving you a result you can use directly in addition, subtraction and other standard algebra. You enter two values: the modulus r, which is the distance of the number from the origin, and the angle theta in degrees, which is measured from the positive real axis. The calculator applies the standard conversion, multiplying the modulus by the cosine of the angle to find the real part a and by the sine of the angle to find the imaginary part b, and returns three results: the real part a, the imaginary part b, and the combined rectangular form written as a + bi. This matters because polar form, r and theta, is the more convenient way to multiply, divide and raise complex numbers to a power, while rectangular form is what you need whenever you are adding or subtracting complex numbers, plotting them, or working with real and imaginary parts separately. Simply type in your modulus and angle in degrees and the real part, imaginary part and rectangular form update immediately. Because results are rounded for display, treat very small nonzero values as effectively zero, and remember the angle must be entered in degrees, not radians, since the calculator converts to radians internally before applying the cosine and sine functions.

Modulus r

Angle theta (degrees)

a = 3

Imaginary part b4

Rectangular form3 + 4i

The formula

From polar form r and theta, the rectangular parts are a = r cos(theta) and b = r sin(theta), giving the complex number a + bi. This is the reverse of taking the modulus and argument.

Worked example

For modulus 5 and angle 53.13 degrees, a = 5 cos(53.13) = 3 and b = 5 sin(53.13) = 4, so the number is 3 + 4i. Enter 5 and 53.130102 to confirm.

Frequently asked questions

How do I convert polar to rectangular?

Multiply the modulus by the cosine of the angle for the real part, and by the sine for the imaginary part.

Why convert to rectangular?

Rectangular form a + bi is needed to add or subtract complex numbers; polar form is better for multiplying and powers.

What units is the angle in?

This calculator takes the angle in degrees and converts internally to radians.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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