Derivatives of Trigonometric Functions
Understanding how trigonometric functions change — their rates of change — is one of the most powerful tools in calculus. These derivatives appear everywhere: physics, engineering, signal processing, and beyond.
Why This Matters
If $y = \sin(x)$ describes the height of a wave at position $x$, then $\frac{dy}{dx}$ tells us the slope of that wave — how steeply it's rising or falling at any point. The derivative captures the instantaneous rate of change.
Foundational Limits
Before deriving the formulas, we need two critical limits:
$$\lim_{h \to 0} \frac{\sin h}{h} = 1 \qquad \text{and} \qquad \lim_{h \to 0} \frac{\cos h - 1}{h} = 0$$
These limits (proven geometrically using the squeeze theorem) are the building blocks for every trig derivative.