No. Trig functions can be defined independently of geometry. One way is by considering the differential equation above. Another is through the Taylor expansion

No. Trig functions can be defined independently of geometry. One way is by considering the differential equation above. Another is through the Taylor expansion of the sin(x), which is periodic as a completely emergent property. Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.