Euler's Formula for the complex exponential

Let's derive Euler's formula e^(it) = cos(t) + i sin(t) and look at related properties. We finish by using this formula to prove the double angle identity from trigonometry. We start by setting x = it in the power series for e to the x, simplify the powers of i, and then group the terms into real and imaginary parts. When we do that, we recognize the patterns for cosine and sine. So we arrive at the expression e to the it equals cos(t) plus i times sin(t), which is Euler’s formula. From there, we use the formula to understand e to the negative it, and we see it’s just the complex conjugate of e to the it. That leads us to a geometric interpretation: both e^(it) and e^(-it) lie on the unit circle in the complex plane, with modulus one. That naturally brings us to Euler’s identity: e^(i*pi) +e 1 = 0, which connects five key mathematical constants in a single elegant equation. We also explore what happens when we raise e to a general complex number like a + i*b. We break it into a real part and a complex exponential part, and find that this complex number lies on a circle of radius e^a. Finally, we use Euler’s formula to derive the double angle identities for sine and cosine. We square e to the it two different ways—once by expanding and once using exponent rules—and then match the real and imaginary parts. This gives us formulas like cosine of 2t equals cosine squared minus sine squared, and sine of 2t equals 2 times sine times cosine. #mathematics #math #complexnumbers #complexanalysis #eulersformula #mathtutorial #taylorseries This video is part of my full Single Variable Calculus II course playlist (Calc 2, MA 241 at NC State University):    • Single Variable Calculus II - Complete Sem...   #calculus2 #singlevariablecalculus