The trigonometric functions and
appear as solutions to the initial value problem for
: (see alternative presentation)
(1)
(2)
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.
Check out this more detailed presentation.
To do:
- 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.