1. What is Trigonometric Form?

The trigonometric form (also called polar form) represents a complex number using its magnitude (r) and angle (θ). It's expressed as \( z = r (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( θ \) is the angle in radians.

2. How Does the Calculator Work?

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

Where:

• \( a \) — Real part
• \( b \) — Imaginary part
• \( r = \sqrt{a^2 + b^2} \) — Magnitude
• \( \theta = \tan^{-1}(b/a) \) — Angle (adjusted for quadrant)

3. Importance of Trigonometric Form

Details: Trigonometric form is essential for multiplying/dividing complex numbers, finding roots/powers (De Moivre's Theorem), and analyzing periodic phenomena in engineering and physics.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will provide both the standard trigonometric form and a version formatted for easy input into Desmos graphing calculator.

5. Frequently Asked Questions (FAQ)

Q1: Why use trigonometric form?
A: It simplifies multiplication/division of complex numbers and makes powers/roots easier to compute using De Moivre's Theorem.

Q2: How is the angle θ determined?
A: We use atan2(b,a) which correctly identifies the angle's quadrant based on the signs of a and b.

Q3: Can I use degrees instead of radians?
A: The calculator uses radians, but you can convert by multiplying by 180/π (≈57.2958).

Q4: What if my complex number is zero?
A: The zero complex number (0 + 0i) has undefined angle and zero magnitude.

Q5: How do I use this in Desmos?
A: Copy the "Desmos Format" result and paste it directly into a Desmos expression line.