Complex Analysis, Video #17 (Complex Arithmetic, Part 17). Applications of De Moivre's and Euler's Formulas to Trigonometric Identities and Calculation of Integrals.
Details: The trig identities derived in previous videos are true, but perhaps not very useful. In this video, we do an application that is useful for (real) integration. Integrate (cos(theta))^3 by using the fact that cos(theta) = (e^(i*theta)+e(-i*theta))/2, the binomial theorem, and then Euler's formula (in your head). Do the integral. Check it graphically on Mathematica (use Mathematica's "Plot" command...as well as the formatting option "PlotStyle").