Trigonometric Functions: cos(t) and sin(t)

The trigonometric functions x=cos(t) and y=sin(t) appear as solutions to the initial value problem for t>0:

  • \frac{dx}{dt} = -y         (1)
  • \frac{dy}{dt} = x           (2)

or in time stepping form

  • x=x-y*dt
  • y=y+x*dt

with the following initial values for t=0:

  • x=\cos(0)=1 and y=\sin(0)=0.

Watch and play with the code.

Understand that by construction with D=\frac{d}{dt}

  • D\cos(t) = -\sin(t)
  • D\sin(t) = \cos(t)

Observe that the differential equations express that the velocity vector (vx,vy)=(\frac{dx}{dt},\frac{dy}{dt}) and the position vector (x,y) satisfies

  • vx* x + vy*y = -y*x + x*y =0,        (velocity is orthogonal to position)

which means that the point (x,y) moves around a circle with radius 1, which give \cos(t) and \sin(t) a geometric meaning with t appearing as an angle. Find out the details of the geometry.

Check out this more detailed presentation.

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