My case for trigonometry: We want to people understand social cycles. For example, heroin becomes fashionable among young people because it feels good. Time goes by and problems emerge with tolerance, addiction, and overdose. The next cohort of young people see what happened to aunts and uncles etc, and give heroin a miss. The cohort after that see their aunts and uncles living clean lives, lives that give no warning. They experiment and find that heroin feels good. The cycle repeats.
These cycles can arise because the fixed points of the dynamics are unstable. The classic simple example uses a second order linear differential equation as a model with a solution such as $e^{at} \sin kt$. We really want people to have some sense of cycles arising from instabilities without anyone driving them. We probably cannot give simple examples of what we mean with trigonometric functions.
I’d say that this is a better argument for calculus and PDEs than trigonometry- the sine function can be defined purely from a calculus point-of-view, and that definition is more similar to what you describe than the trigonometry perspective
My case for trigonometry: We want to people understand social cycles. For example, heroin becomes fashionable among young people because it feels good. Time goes by and problems emerge with tolerance, addiction, and overdose. The next cohort of young people see what happened to aunts and uncles etc, and give heroin a miss. The cohort after that see their aunts and uncles living clean lives, lives that give no warning. They experiment and find that heroin feels good. The cycle repeats.
These cycles can arise because the fixed points of the dynamics are unstable. The classic simple example uses a second order linear differential equation as a model with a solution such as $e^{at} \sin kt$. We really want people to have some sense of cycles arising from instabilities without anyone driving them. We probably cannot give simple examples of what we mean with trigonometric functions.
I’d say that this is a better argument for calculus and PDEs than trigonometry- the sine function can be defined purely from a calculus point-of-view, and that definition is more similar to what you describe than the trigonometry perspective