Complex Number to Trigonometric Form Calculator

The trigonometric form of a complex number z reads

z = r × [cos(φ) + i × sin(φ)],

where:

• r is the modulus, i.e., the distance from (0,0) to z; and
• φ is the argument, i.e., the angle between the x-axis and the radius between (0,0) and z.

You can see r and φ in the picture below. Using basic trigonometry, we see that
cos(φ) = a × r and sin(φ) = b × r, exactly as claimed by the formula above.

To convert a complex number z = a + bi from rectangular to trigonometric form, you need to determine both the magnitude and the argument of z:

1. Compute the magnitude as r = √(a² + b²).
2. Compute the phase as φ = atan(b / a). If needed, correct by ±π to end up in the correct quadrant of the plane.
3. Write down the trigonometric form as z = r × [cos(φ) + i × sin(φ)].

To use the power of our tool, you need to feed it with the real and imaginary parts of your complex number, so with a and b from the a + bi form.

Our complex number to trigonometric form calculator will immediately compute the magnitude r and phase (argument) φ of your number and then display the trigonometric form in two ways: using radians and degrees.

🙋 For some particular complex numbers you will get a third form as well: it will use π × rad as the angle unit. In this way, angles like π/4 have a chance of appearing.

Satisfied with the complex number to trigonometric form calculator? features a rich collection of calculators dedicated to complex numbers! Here's a full list in case you want to learn more:

• Complex number calculator;
• a+bi form calculator;
• Multiply complex numbers calculator;
• Divide complex numbers calculator;
• Imaginary number calculator;
• Complex number to polar form calculator;
• Complex number to rectangular form calculator;
• i calculator; and
• Polar form calculator.