To take the shortest path to Calculus, we don't need to cover all the trigonometric identities. We just need to

  1. define the sin & cos function in terms of a right triangle
  2. teach the unit circle coordinate values of simple angles:
    0, pi/6, pi/4, pi/3, pi/2, ...

Basic sin & cos can form the rotation matrix, which will later lead to hyperbolic rotations of Special Relativity.