Complex Number To Trigonometric Form
Convert complex numbers a+bi to trigonometric form r(cos theta + i sin theta). Enter real and imaginary parts to find magnitude r = sqrt(a^2+b^2) and angle theta in degrees.
About This Calculator
Complex Number to Trigonometric Form Conversion
Every complex number z = a + bi (rectangular form) can also be expressed in trigonometric form as r(cos theta + i·sin theta), where r is the magnitude and theta is the angle.
The conversion formulas are:
- r = sqrt(a^2 + b^2) -- magnitude (modulus)
- theta = arctan(b / a) -- angle (argument) in degrees
The trigonometric form is also equivalent to the polar form r·e^(itheta) via Euler's formula.
Example
The complex number 3 + 4i converts to trigonometric form as 5(cos 53.13 deg + i·sin 53.13 deg) because:
r = sqrt(3^2 + 4^2) = 5, theta = arctan(4/3) = 53.13 deg
Trigonometric form is especially useful for multiplying complex numbers (multiply magnitudes, add angles) and for representing oscillations, waves, and alternating current in physics and engineering.
Frequently Asked Questions
How do you convert a complex number to trigonometric form?
For a complex number a+bi, calculate magnitude r = sqrt(a^2+b^2) and angle theta = arctan(b/a). The trigonometric form is r(cos theta + i·sin theta). For example, 3+4i becomes 5(cos 53.13 deg + i·sin 53.13 deg).
What is the trigonometric form of a complex number?
The trigonometric form (also called polar form) represents a complex number as r(cos theta + i·sin theta), where r is the magnitude (distance from origin) and theta is the angle from the positive real axis measured in degrees or radians.
How is trigonometric form different from polar form?
Trigonometric form r(cos theta + i·sin theta) and polar form r·e^(itheta) are mathematically equivalent via Euler's formula e^(itheta) = cos theta + i·sin theta. Trigonometric form explicitly shows the cosine and sine components, while polar form uses the exponential notation more common in advanced mathematics and engineering.
What if my imaginary part is negative?
Negative imaginary parts work correctly. The calculator uses atan2(b, a) which handles all four quadrants and returns the correct angle between -180 deg and 180 deg (or -pi to pi radians).
What is an example of converting to trigonometric form?
For the complex number 1 + i: magnitude r = sqrt(1^2+1^2) = sqrt2 ≈ 1.414, angle theta = arctan(1/1) = 45 deg. The trigonometric form is sqrt2(cos 45 deg + i·sin 45 deg). For 0 + 3i (pure imaginary): r = 3, theta = 90 deg, giving 3(cos 90 deg + i·sin 90 deg).
Is this complex number to trigonometric form calculator free?
Yes, this complex number to trigonometric form converter is completely free to use with no registration, login, or usage limits.
Why use trigonometric form for complex numbers?
Trigonometric form simplifies multiplication, division, and exponentiation of complex numbers. It is essential in electrical engineering (AC circuit phasors), signal processing (Fourier transforms), quantum mechanics, and control systems where phase and magnitude relationships are fundamental.