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 the trigonometric form equation:

Where:

• \( r \) — Magnitude (distance from origin)
• \( \theta \) — Angle with positive real axis (in radians)
• \( i \) — Imaginary unit (\( \sqrt{-1} \))

Conversion to Rectangular Form: The calculator also shows the standard form \( a + bi \) where:

• \( a = r \cos \theta \) (real part)
• \( b = r \sin \theta \) (imaginary part)

3. Importance of Trigonometric Form

Details: The 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 magnitude (must be ≥0) and angle in radians. The calculator will show both the trigonometric form and the equivalent rectangular form.

5. Frequently Asked Questions (FAQ)

Q1: How do I convert degrees to radians?
A: Multiply degrees by π/180. Many calculators have a degree-to-radian conversion function.

Q2: What's the range for θ?
A: Typically between -π and π, but any real number is valid due to periodicity of trigonometric functions.

Q3: Can I use negative magnitude?
A: No, magnitude is always non-negative. Negative values will be converted to positive.

Q4: What's Euler's formula relation?
A: The trigonometric form can be written as \( z = re^{i\theta} \) using Euler's formula.

Q5: When is trigonometric form most useful?
A: Particularly helpful for powers and roots of complex numbers (De Moivre's Theorem).