Complex Number to Trigonometric Form Calculator
Convert complex numbers from rectangular form (a + bi) to trigonometric form r[cos(φ) + isin(φ)]
Convert Rectangular to Trigonometric Form
Horizontal component on complex plane
Vertical component on complex plane
Current Rectangular Form
Trigonometric Form Results
Conversion formulas:
• Magnitude: |z| = √(a² + b²) = √(0² + (0)²) = 0.0000
• Argument: φ = atan2(b, a) = atan2(0, 0) = 0.0000 rad
• Trigonometric form: z = |z| × [cos(φ) + i × sin(φ)]
Common Example Conversions
Complex Number Forms
Rectangular Form
z = a + bi
Where a = real part, b = imaginary part
Trigonometric Form
z = r[cos(φ) + isin(φ)]
Where r = magnitude, φ = argument
Polar (Exponential) Form
z = r × e^(iφ)
Compact exponential notation
Conversion Formulas
Rectangular → Trigonometric
r = √(a² + b²)
φ = atan2(b, a)
z = r[cos(φ) + isin(φ)]
Trigonometric → Rectangular
a = r × cos(φ)
b = r × sin(φ)
z = a + bi
Common Angles
Quick Tips
Trigonometric form is useful for multiplication and division
Magnitude r is always non-negative
Argument φ is typically given in [-π, π]
Use atan2(b, a) for correct quadrant determination
Understanding Trigonometric Form of Complex Numbers
What is Trigonometric Form?
The trigonometric form represents a complex number as z = r[cos(φ) + isin(φ)], where 'r' is the magnitude (distance from origin) and 'φ' is the argument (angle from positive real axis).
Why Use Trigonometric Form?
- •Multiplication and division become easier
- •Powers and roots are simplified using De Moivre's theorem
- •Clear geometric interpretation on complex plane
- •Connection between algebra and trigonometry
Conversion Process
From: z = a + bi
To: z = r[cos(φ) + isin(φ)]
Step 1: Calculate magnitude using r = √(a² + b²)
Step 2: Calculate argument using φ = atan2(b, a)
Step 3: Write trigonometric form
Note: atan2(b, a) automatically handles quadrant determination correctly!
Operations with Trigonometric Form
Multiplication
z₁ × z₂ = r₁r₂[cos(φ₁ + φ₂) + isin(φ₁ + φ₂)]
Multiply magnitudes, add arguments
Division
z₁ ÷ z₂ = (r₁/r₂)[cos(φ₁ - φ₂) + isin(φ₁ - φ₂)]
Divide magnitudes, subtract arguments
Powers (De Moivre's Theorem)
z^n = r^n[cos(nφ) + isin(nφ)]
Raise magnitude to power n, multiply argument by n
nth Roots
z^(1/n) = r^(1/n)[cos((φ + 2kπ)/n) + isin((φ + 2kπ)/n)]
Where k = 0, 1, 2, ..., n-1