# derivatives of sine and cosine

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

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

as  $t\to x$.  Here we used the known limit  $\displaystyle\lim_{u\to 0}\frac{\sin{u}}{u}=1$  (see this entry (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0)).

The derivative of cosine is calculated similarly:

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

as  $t\to x$.

Title derivatives of sine and cosine DerivativesOfSineAndCosine 2013-03-22 16:58:58 2013-03-22 16:58:58 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Derivation msc 26A09 DerivativesOfSinXAndCosX LimitOfDisplaystyleFracsinXxAsXApproaches0 DefinitionsInTrigonometry LimitRulesOfFunctions