Why d/dx(sin x) = cos x? Use the angle-sum formula and two key limits: lim(h→0) sin(h)/h = 1 and lim(h→0) (cos h − 1)/h = 0.
Walk through
Step 1 of 5
Start from the definition
f′(x) = lim(h→0) [sin(x + h) − sin x] / h.
The two key limits
These come from the squeeze theorem and the unit circle — everything else follows.
Your turn
Adapt the same argument to show d/dx(cos x) = −sin x.
Both key limits require angles in radians. In degrees, lim sin(h)/h = π/180 — which is exactly why calculus uses radians everywhere.