How do you know where a sine function is increasing or decreasing?
Thanks!
But note that $\displaystyle f^{\prime}\!\left(x\right)>0$ doesn't necessarily mean that $\displaystyle f\!\left(x\right)>0$ and that $\displaystyle f^{\prime}\!\left(x\right)<0$ doesn't necessarily mean that $\displaystyle f\!\left(x\right)<0$ In the case of $\displaystyle f^{\prime}\!\left(x\right)=\cos x$, there will be two intervals where $\displaystyle f^{\prime}\!\left(x\right)<0$ but $\displaystyle f\!\left(x\right)>0$ and $\displaystyle f^{\prime}\!\left(x\right)>0$ but $\displaystyle f\!\left(x\right)<0$
So what intervals is $\displaystyle \cos x<0$ and $\displaystyle \cos x>0$?
Why don't you tell us what question you are REALLY asking? You originally asked "How do you know where a sine function is increasing or decreasing?" and that was interpreted as a function that involves sine in some way. Are you asking specifically about sin(x)? Or about $\displaystyle sin(\omega(x+ \Omega))$?