# Trigonometry/Derivative of Sine

To find the derivative of sin(θ).

$\frac{d}{d\theta} \sin(\theta) = \lim_{h \rightarrow 0} \frac{\sin(\theta+h)-\sin(\theta)}{h} = \lim_{h \rightarrow 0} \frac{2\cos(\theta+\frac{h}{2})\sin(\frac{h}{2})}{h} = \lim_{h \rightarrow 0} {\cos\left(\theta+\frac{h}{2}\right)}\frac{\sin\left(\frac{h}{2}\right)}{\frac{h}{2}}$.

Clearly, the limit of the first term is $\cos(\theta)$ since $\cos(\theta)$ is a continuous function. Write k = h2; the second term is then

$\frac{\sin(k)}{k}$.

Which we proved earlier tends to 1 as $k \rightarrow 0$.

And since

$k \rightarrow 0 \text { as } h \rightarrow 0$,

the limit of the second term is 1 too. Thus

$\frac{d}{d\theta} \sin(\theta) = \cos(\theta)$.