Calculus students who know the quotient rule and the derivatives of e^x, cos x, sin x are ready to begin understanding Euler’s formula which connects the 3 functions and the imaginary number i. It’s a deep formula, and this exploration only scratches the surface, but this should give students a glimpse of the powerful interconnectedness of different branches of math, enticing them to go on to complex analysis after vector calc.
The first page of this pdf file lets them use their knowledge of derivatives to work out the truth of Euler’s identity. Then the second page uses Euler’s identity to give them access to the double angle formulas. If you like, you can extend this to triple angle, quadruple angle, n-tuple angle identities. With this proof and derivation of double angle identities, calculus students then have a firmer grasp of their trig knowledge.
Hat tip to Robbie Martinez for the proof on the first page, and to Paul Zeitz & Tristan Needham whose books helped me understand the second page.