Derivatives of Trig Functions

The derivatives of the trig functions simply tell you how fast sine, cosine and the rest are changing at each angle. They matter because anything that swings back and forth, like a spring, a pendulum or AC current, is described by sines and cosines, and its velocity is their derivative. The two you really need are $\frac{d}{dx}\sin x = \cos x$ (the slope of the sine wave is the cosine wave) and $\frac{d}{dx}\cos x = -\sin x$ (the slope of the cosine wave is an upside-down sine wave). The longer list is just those two run through the quotient rule, so don't let it intimidate you.

Picture it first. Before any algebra, think about the shape of the sine wave. At $x = 0$ it's climbing as steeply as it ever does, with slope $1$, and $\cos 0 = 1$. At the top of the wave, $x = \frac\pi2$, it's momentarily flat, slope $0$, and $\cos\frac\pi2 = 0$. At $x = \pi$ it's falling as fast as it can, slope $-1$, and $\cos\pi = -1$. The slopes of sine are tracing out cosine.

Where $\cos x$ comes from. To confirm it, go back to the definition. Using the angle-sum formula,

As $h \to 0$, $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$, leaving $\cos x$. Those two limits are only true when angles are measured in radians. In degrees you'd get an annoying factor of $\frac{\pi}{180}$ everywhere, which is why calculus always uses radians.

The others. Every other trig function is a quotient of sine and cosine, so the quotient rule does the rest. For example, $\tan x = \frac{\sin x}{\cos x}$, so its derivative is $\frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \sec^2 x$. A pattern helps you keep them straight: the "co-" functions ($\cos$, $\csc$, $\cot$) are the ones whose derivatives get a minus sign.

Anything oscillating, from a mass on a spring to AC current, has a velocity or current that is a derivative of a sine or cosine, shifted a quarter period ahead.