The trigonometric functions and appear as solutions to the initial value problem for :
or in time stepping form
with the following initial values for :
- and .
Understand that by construction with
Observe that the differential equations 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 .
- 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
- , ,
where the acceleration is balanced by a spring force scaling with position measuring the extension of the spring.