Euler's Formula & Identity

Connect exponentiation and trigonometry. Euler's Formula, $e^{i\theta} = \cos\theta + i\sin\theta$, maps circular coordinates in the complex plane, yielding the famous identity $e^{i\pi} + 1 = 0$.

Interactive Sandbox

Euler's Complex Plane

Drag the vector on the circle directly or adjust the angle slider. Witness the real component (cosine) and imaginary component (sine) project onto the axes. Increase Taylor terms to see the spiral converge!

Unit Circle Plot: z = e^(iθ) = cos(θ) + i sin(θ)
Taylor Series Spiral: Σ (iθ)^k / k!

Active Complex Value

e^(iθ) = cos(0) + i sin(0) = 1.00 + 0.00i
Angle (θ in Radians) 0.00 rad
Taylor Series Terms (N) 6
Drag the glowing vector node directly around the unit circle to see its coordinates sweep.
Algebraic Solver

Euler's & Taylor expansion Steps

Trace the numeric evaluation of Euler's formula and observe how imaginary terms group dynamically to form the circular spiral.

Numeric Substitution & Evaluation


Euler's Identity Derived!

When θ = π (180°), we have:
eiπ = cos(π) + i sin(π) = -1 + 0i = -1.
Adding 1 to both sides yields: eiπ + 1 = 0.

Approximation Details


Real Component (cos θ)

cos(0.00) = 1.00

Imaginary Component (sin θ)

sin(0.00) = 0.00

Taylor Approximation Tip

z_approx = 1.00 + 0.00i

Evaluation

Euler's Identity Quiz

Test your understanding of complex coordinates, Taylor expansions, and identity relations.

Question 1 of 5 Score: 0/0

What is the complex value of e^(iπ/2)?

Correct Answer!

Explanation text...

Euler & Complex Plane Cheat Sheet


  • Euler's Formula: eiθ = cos(θ) + i sin(θ)
  • Real Part (cos θ) = Re(z)
  • Imaginary Part (sin θ) = Im(z)
  • Euler's Identity: eiπ + 1 = 0