Complex Number Trigonometric Form Calculator with Answers

Complex Number Trigonometric Form:

\[ z = r (\cos \theta + i \sin \theta) \] \[ r = \sqrt{a^2 + b^2} \] \[ \theta = \arctan\left(\frac{b}{a}\right) \]

Magnitude (r):

Angle (θ):

Trigonometric Form:

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1. What is Trigonometric Form of Complex Numbers?

The trigonometric form (also called polar form) represents a complex number using its magnitude and angle. It's particularly useful for multiplication, division, and exponentiation of complex numbers.

2. How Does the Calculator Work?

The calculator uses these formulas:

\[ r = \sqrt{a^2 + b^2} \] \[ \theta = \arctan\left(\frac{b}{a}\right) \] \[ z = r (\cos \theta + i \sin \theta) \]

Where:

\( a \) — Real part of complex number (unitless)
\( b \) — Imaginary part of complex number (unitless)
\( r \) — Magnitude (always non-negative, unitless)
\( \theta \) — Angle in radians (-π to π)
\( i \) — Imaginary unit (√-1)

Explanation: The calculator first computes the magnitude (distance from origin) and then determines the angle while accounting for the correct quadrant.

3. Importance of Trigonometric Form

Details: Trigonometric form simplifies complex number operations. Multiplication becomes multiplying magnitudes and adding angles, while division becomes dividing magnitudes and subtracting angles.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will provide the magnitude, angle (in radians), and full trigonometric form.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from rectangular form?
A: Rectangular form uses a + bi, while trigonometric form uses magnitude and angle, making certain operations easier.

Q2: What's the range for the angle θ?
A: The angle is typically between -π and π radians (-180° to 180°), known as the principal value.

Q3: How do I convert back to rectangular form?
A: Use a = r cosθ and b = r sinθ to get back to a + bi form.

Q4: What if my complex number is zero?
A: For z = 0, r = 0 and θ is undefined (calculator will show an error).

Q5: Why use radians instead of degrees?
A: Radians are the standard unit in higher mathematics and most programming languages, though you can convert if needed.