Defining the sine function as an oscillator

When I do something trigonometric, I reach for a library implementing common trig functions, e.g. Math.sin. From school through university through work, I’ve been taught to treat these functions as black boxes. Before we had calculators at school, there were books of sine tables - more black boxes. How do these functions actually work? How do you define sin(x)?

function sin(x) {
  return /* ??? */;
}

One definition of sine which I was familiar with is: sine is the function traced out by a point on a rotating circle, when viewing the circle from the edge. If you try to implementing sine with this definition, you look up the definition of a circle, plug in the angles, and get ...

function sin(x) {
  return Math.sin(x);
}

... great. But a different definition of sine is: sine is the function traced out by an object on a spring. A spring exerts a force on the object, pushing it back towards the equilibrium. An ideal spring is a “simple harmonic oscillator”, which means that it exerts a force proportional to the distance from the equilibrium.

Based on this definition, we can write a sin function which works by simulating a spring:

var delta = 0.001;
function springSin(x) {
  var velocity = delta;
  var y = 0;
  for (var t = 0; t < x; t += delta) {
    y += velocity;
    velocity -= y * (delta * delta);
  }
  return y;
}

We can optimize this using the repeating and symmetric nature of the sine function:

var delta = 0.001;
function optimizedSpringSin(x) {
  var sign = 1;
  if (x < 0) { x = -1; sign = -sign; }  // [0, inf]
  x = x % (2 * Math.PI); // [0, 2*Math.PI]
  if (Math.PI < x) { x -= Math.PI; sign = -sign; } // [0, Math.PI]
  if (Math.PI/2 < x) { x = Math.PI - x; } // [0, 1/2 Math.PI]
  // Now x is in [0, 1/2 PI]
  return springSin(x) * sign;
}

Here’s a plot of the Math.sin function (green) next to the optimizedSpringSin function (black, slightly offset):

Why does the iterative springSin function approximate the true sine function? Because there’s a relationship between oscillation and circles/triangles. Unfortunately I don’t understand that relationship.

The actual implementation of sin in math libraries uses a “Taylor series approximation” of the sine function. Unfortunately I don’t understand that, either.

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