Free Complex Number to Trigonometric Form Calculator

This rectangular to trigonometric form calculator is a free online tool that converts a complex number from its standard $a+bi$ representation into the equivalent trigonometric (polar) form. By computing both the magnitude and argument of the number, it provides the expression $r(\cos\varphi + i\sin\varphi)$ — a format that is indispensable for understanding the geometry of complex numbers and performing operations like multiplication, division, and exponentiation.

What Is the Trigonometric Form of a Complex Number?

Every complex number $z = a + bi$ (with real part $a$ and imaginary part $b$) corresponds to a point $(a,b)$ in the complex plane. Its trigonometric form is:

$$z = r \bigl( \cos\varphi + i \sin\varphi \bigr)$$

where:

• $r = \sqrt{a^{2} + b^{2}}$ is the modulus (the Euclidean distance from the origin to the point).
• $\varphi$ is the argument (the angle measured counter‑clockwise from the positive real axis to the radius vector).

The relations $\cos\varphi = a/r$ and $\sin\varphi = b/r$ follow directly from the geometry of a right triangle with legs $a$ and $b$ and hypotenuse $r$.

How to Convert from Rectangular to Trigonometric Form Manually

If you want to perform the conversion by hand, follow these steps:

1. Calculate the modulus

   $$r = \sqrt{a^{2} + b^{2}}.$$

2. Determine the argument
   Use the arctangent function, but be careful with the quadrant:

   • If $a > 0$: $\varphi = \arctan(b/a)$.
   • If $a < 0$ and $b \ge 0$: $\varphi = \arctan(b/a) + \pi$.
   • If $a < 0$ and $b < 0$: $\varphi = \arctan(b/a) - \pi$.
   • Special cases: $a = 0$ and $b > 0 \implies \varphi = \pi/2$; $a = 0$ and $b < 0 \implies \varphi = -\pi/2$.

   The result is usually given as the principal argument in the interval $(-\pi, \pi]$ (or $[0, 2\pi)$ in some conventions).

3. Write the trigonometric form

   $$z = r \bigl( \cos\varphi + i \sin\varphi \bigr).$$

Example: Convert $z = -1 + i$

• Modulus: $r = \sqrt{(-1)^{2} + 1^{2}} = \sqrt{2}$.
• Argument: $a=-1<0$ and $b=1>0$ → second quadrant. The reference angle is $\pi/4$. Adding $\pi$ gives $\varphi = 3\pi/4$.
• Result: $$z = \sqrt{2} \left( \cos\!\Bigl(\frac{3\pi}{4}\Bigr) + i \sin\!\Bigl(\frac{3\pi}{4}\Bigr) \right).$$

Using the Online Complex Number to Trigonometric Form Calculator

With this complex number magnitude and argument calculator, the process is fully automated:

1. Enter the real part $a$ and the imaginary part $b$ in the corresponding input fields.
2. The tool instantly computes $r$ and $\varphi$ using the formulas above, including the automatic quadrant correction.
3. The trigonometric form is displayed in two common angle formats: radians (decimal) and degrees. Additionally, when the argument is a rational multiple of $\pi$, an exact representation (e.g., $\frac{3\pi}{4}$) is shown.

The calculator is designed for anyone who needs fast, accurate conversions — from students studying algebra to professionals working with phasors in electrical engineering.

Why Use the Trigonometric Form?

The trigonometric (or polar) representation simplifies many operations:

• Multiplication: $r_{1}\bigl(\cos\varphi_{1}+i\sin\varphi_{1}\bigr) \cdot r_{2}\bigl(\cos\varphi_{2}+i\sin\varphi_{2}\bigr) = r_{1}r_{2}\bigl(\cos(\varphi_{1}+\varphi_{2})+i\sin(\varphi_{1}+\varphi_{2})\bigr)$.
• Division: The moduli divide and the arguments subtract.
• Powers: De Moivre’s theorem states that $\bigl(r(\cos\varphi+i\sin\varphi)\bigr)^{n} = r^{n}\bigl(\cos(n\varphi)+i\sin(n\varphi)\bigr)$.

Thus, converting to trig form turns complex arithmetic into simple operations on magnitudes and angles.

Getting Started

Whether you call it the trigonometric form, polar form, or the magnitude‑argument representation, this free online converter gives you a clear, step‑by‑step result for any complex number. It is the ideal companion for homework, exam preparation, or engineering calculations.