Complex-plane identities and Euler notation

Connect complex numbers, rotations and trigonometric identities in this precalculus extension practice. Euler’s formula is supplied as a bridge result, while the triple-angle identities are derived algebraically from complex powers. Questions cover real and imaginary parts, conjugation, unit-complex transformations and exact trigonometric deductions. Both question formats include detailed explanations and proof-error analysis. No calculus or analytic proof of the complex exponential is assumed; the goal is to understand what the notation means and justify the algebra that follows.