Complex Number to Trigonometric Form Calculator

This complex number to trigonometric form calculator will help if you're still not quite sure how to find the trigonometric form of a complex number a + bi. In the brief article below, we explain what the trig form of a complex number is and how to quickly convert from rectangular to trigonometric form.

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.

Satisfied with the complex number to trigonometric form calculator? Omni 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.

The answer is i = √2 × [cos(π/4) + i × sin(π/4)]. To arrive at this result, observe that the magnitude of i equals r = √(1² + 1²) = √2. Then observe that the argument of i equals π/4 since atan(1/1) = π/4.

If you have a complex number in the polar form r × exp(iφ), it suffices to take r and φ from the polar form and write down r × [cos(φ) + i × sin(φ)] to get the trig form of your complex number. No calculations required!