AI Tutor
If you track the slope of a rolling sine wave at every peak, valley, and crossing, the resulting slope graph forms a perfect cosine wave.
To prove this link rigorously, we use the formal definition of a derivative: fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
Before expanding the difference quotient, what trigonometric identities and small-angle limits will unlock the algebra?

Essential Trigonometric Tools

The angle addition formulas decompose shifted trigonometric functions into products of individual angles: sin⁑(x+h)=sin⁑(x)cos⁑(h)+cos⁑(x)sin⁑(h)\sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h) and cos⁑(x+h)=cos⁑(x)cos⁑(h)βˆ’sin⁑(x)sin⁑(h)\cos(x+h) = \cos(x)\cos(h) - \sin(x)\sin(h).
We also need two fundamental trigonometric limits as angle hh (measured strictly in radians) shrinks to zero: lim⁑hβ†’0sin⁑(h)h=1\lim_{h \to 0} \frac{\sin(h)}{h} = 1 and lim⁑hβ†’0cos⁑(h)βˆ’1h=0\lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0.
Now that our identities and limits are ready, what happens when we substitute f(x)=sin⁑(x)f(x) = \sin(x) directly into the limit definition?

Proof: Derivative of Sine

Let f(x)=sin⁑(x)f(x) = \sin(x). Writing out the limit of the difference quotient gives:
fβ€²(x)=lim⁑hβ†’0sin⁑(x+h)βˆ’sin⁑(x)hf'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}
Expanding sin⁑(x+h)\sin(x+h) using our angle addition formula yields: