1. What is Trigonometric Form?

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

2. How Does the Calculator Work?

The calculator converts from rectangular form (a + bi) to trigonometric form:

Where:

• \( r = \sqrt{a^2 + b^2} \) — Magnitude (modulus)
• \( \theta = \arctan\left(\frac{b}{a}\right) \) — Angle (argument) in radians
• \( a \) — Real part
• \( b \) — Imaginary part
• \( i \) — Imaginary unit

Explanation: The calculator computes the magnitude using the Pythagorean theorem and the angle using the arctangent function (atan2 for correct quadrant determination).

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 compute the magnitude (r), angle (θ in radians), and display the trigonometric form.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from rectangular form?
A: Rectangular form (a + bi) uses Cartesian coordinates, while trigonometric form uses polar coordinates (magnitude and angle).

Q2: Can I get the angle in degrees?
A: Multiply the radians value by (180/π) to convert to degrees. The calculator shows radians as it's the standard unit in mathematics.

Q3: What if my complex number is zero?
A: The magnitude would be zero and the angle undefined, as 0 has no meaningful direction.

Q4: Why use trigonometric form?
A: It's particularly useful for powers and roots of complex numbers (De Moivre's Theorem) and in electrical engineering (phasors).

Q5: How is this related to Euler's formula?
A: Euler's formula \( e^{i\theta} = \cos \theta + i \sin \theta \) allows writing the trigonometric form more compactly as \( z = re^{i\theta} \).