The trigonometric functions and appear as solutions to the initial value problem for : (see alternative presentation)
or in time stepping code form
- (as )
- (as )
with the following initial values for :
- and .
Understand that by construction with
Conclude that both cos(t) and sin (t) solves the second-order differential equation (initial value problem) in a function u(t)
- for t > 0,
with the following initial values:
Observe that the differential equations (1)-(2) express that the velocity vector and the position vector satisfies
- , (velocity is orthogonal to position)
which means that the point moves around a circle with radius 1, which give and a geometric meaning with t appearing as an angle. Find out the details of the geometry.
- Compute Pi (3.14….) by computing the first value of t > 0 such that sin(t)=0 by time stepping (1)-(2).
- Show that by construction . Hint: Note that solves with initial conditions and and so is the linear combination .
- Similarly show other properties of and .
- See that (1)-(2) models a harmonic oscillator of a body connected to a fixed point with a spring modeled by dx = – v*dt and dv = x*dt, where the acceleration dv/dt is balanced by a spring force scaling with position x measuring the extension of the spring.