PlanetMath: derivatives of sine and cosine

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derivatives of sine and cosine

(Derivation)

The derivation of the derivatives of sine and cosine is a bit simpler by using the prosthaphaeresis formulas

$\displaystyle \sin\alpha-\sin\beta = \,2\sin \left( \frac{\alpha\!-\!\beta}{2} \right) \,\cos \left( \frac{\alpha\!+\!\beta}{2} \right),$

(1)

$\displaystyle \cos\alpha-\cos\beta = -2\sin \left( \frac{\alpha\!+\!\beta}{2} \right) \, \sin\left( \frac{\alpha\!-\!\beta}{2} \right).$

(2)

Let $x,\,t$ be any real numbers such that $t \neq x$ . Then we obtain

$\displaystyle \frac{\sin{x}-\sin{t}}{x-t} = \frac{2\sin \left( \frac{x-t}{2} \r... ...) \;\; \longrightarrow\; 1\cdot\cos \left( \frac{x\!+\!x}{2} \right) = \cos{x},$

as $t\to x$ . Here we used the known limit $\displaystyle\lim_{u\to0}\frac{\sin{u}}{u} = 1$ (see this entry).

The derivative of cosine is calculated similarly:

$\displaystyle \frac{\cos{x}-\cos{t}}{x-t} = \frac{-2\sin \left( \frac{x+t}{2} \... ...ngrightarrow\; -1 \cdot 1\cdot \sin \left( \frac{x\!+\!x}{2} \right) =-\sin{x},$