Differentiating sin x and cos x

From first principles (using $\sin \theta \approx \theta$ for small $\theta$): the derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$.

The first-principles limit ${\lim }_{h\to 0}\frac{\sin h}{h}=1$ only holds when $h$ is in radians, so these derivatives are only valid in radians.