What is the derivative of sin x with respect to x?

The derivative of sin x is cos x, provided x is measured in radians. The reason radians matter is that the proof rests on the limit of sin x divided by x approaching one as x approaches zero, and that clean result only holds in radian measure. Degrees would drag an awkward constant into every formula, which is exactly why radians are the standard unit in calculus. The result makes sense visually: the derivative is the slope of the sine curve, and the sine graph is rising most steeply as it crosses zero, which is precisely where cosine peaks at one, while sine flattens out at its maximum, where cosine has fallen to zero. Differentiating repeatedly cycles through four functions: sin becomes cos, cos becomes minus sin, minus sin becomes minus cos, and minus cos returns to sin. That four-step cycle is why sine and cosine describe oscillations so naturally, from pendulums to alternating current.

Because the second derivative of sine is its own negative, sine solves the equation that describes simple harmonic motion.