1. What is the Trigonometric Form of Complex Numbers?

The trigonometric form (also called polar form) represents a complex number using its magnitude (r) and angle (θ). This form is particularly useful for multiplication and division of complex numbers, as well as for computing powers and roots.

2. How Does the Calculator Work?

The calculator uses the trigonometric form equation:

Where:

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

Explanation: The calculator converts between trigonometric form and rectangular form (a + bi) using trigonometric functions.

3. Importance of Trigonometric Form

Details: The trigonometric form simplifies complex number operations. Multiplication becomes magnitude multiplication and angle addition, while division becomes magnitude division and angle subtraction.

4. Using the Calculator

Tips: Enter the 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. Most calculators have a degree-to-radian conversion function.

Q2: What's the range for θ?
A: The angle can be any real number, but is typically represented in the range (-π, π] or [0, 2π).

Q3: 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 \( z = re^{i\theta} \).

Q4: Can I represent negative magnitudes?
A: Magnitude is always non-negative. For negative r, add π to θ and use positive r.

Q5: How does this help with complex number operations?
A: To multiply two complex numbers: multiply magnitudes and add angles. To divide: divide magnitudes and subtract angles.