1. What is Trigonometric Form of Complex Numbers?

The trigonometric form (also called polar form) represents a complex number using its magnitude (r) and angle (θ). It's particularly useful in calculus 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)
• \( \theta = \tan^{-1}(b/a) \) — Angle (argument) in radians
• \( a \) — Real part
• \( b \) — Imaginary part
• \( i \) — Imaginary unit

Explanation: The trigonometric form represents the complex number as a point in the complex plane with distance r from the origin and angle θ from the positive real axis.

3. Importance of Trigonometric Form

Details: Trigonometric form simplifies complex number operations - multiplication becomes multiplying magnitudes and adding angles, exponentiation becomes raising the magnitude to the power and multiplying the angle by the exponent (De Moivre's Theorem).

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will compute the magnitude (r), angle (θ), and display the trigonometric form.

5. Frequently Asked Questions (FAQ)

Q1: Why use trigonometric form?
A: It simplifies complex number operations, especially multiplication, division, and exponentiation, which are common in calculus and engineering applications.

Q2: How is θ different in degrees vs radians?
A: This calculator uses radians (standard in calculus). To convert to degrees, multiply by 180/π.

Q3: What's the range for θ?
A: The principal value of θ is between -π and π (-180° to 180°).

Q4: What if my complex number is 0?
A: For z = 0, r = 0 and θ is undefined.

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 as \( z = re^{i\theta} \).