Derivatives (in blue) of some trig functions (in brown).

The figures below show the derivatives with respect to θ of sin(θ), sin(2θ), tan(θ), and sin(θ2), respectively.

Derivative of sin(u) Derivative of sin(2u)

Derivative of tan(u) Derivative of sin(u^2)

Here are the graphs of sin(θ)/θ and (cos(θ)−1)/θ:
Graph of sin(h)/h Graph of (cos(h)-1)/h