Math
Euler's identity in motion
Connect the unit circle, sine and cosine with the vector sum of the complex exponential series and Euler’s identity.
Try this
The left point is the exact complex exponential on the unit circle. The right arrows add its Taylor terms, with a separate scale to fit intermediate sums. Increase the term count and compare the error with the integral-remainder bound.
Equations used in the module
Equations exp(iθ) = cos θ + i sin θ exp(iθ) = Σn≥0 (iθ)ⁿ/n! After N terms, |remainder| ≤ |θ|ᴺ/N! for real θ At θ=π: exp(iπ)+1=0
