# Thread: sine/cosine function -----> are both true for when it is negative?

1. ## sine/cosine function -----> are both true for when it is negative?

Do the following rules properly describe at which intervals both the unrestricted sine and cosine functions are negative? Is only one of them right or both?

1) [Xmin - period/4, Xmin + period/4]
2) [x2 + pn, x1 +pn] (p = period) (n = integers)

I'm trying to find similar points using my calculator with these two rules and I can't seem to find any which leads me to believe, I am doing something wrong or these rules are wrong.

2. Originally Posted by s3a
Do the following rules properly describe at which intervals both the unrestricted sine and cosine functions are negative? Is only one of them right or both?
In the first case, both are negative if $\displaystyle \pi < x < \frac{{3\pi }}{2}$.
So in general, by using equivalent intervals we get for any integer $\displaystyle K$
$\displaystyle \pi + 2K\pi < x < \frac{{3\pi }}{2} + 2K\pi \;\& \;\pi \left( {2K + 1} \right) < x < \pi \left( {\frac{3}{2} + 2K} \right)$.

Post script.
Upon rereading your question, it occurs to me that it may mean this:
“When are $\displaystyle \sin(x)$ and $\displaystyle \cos(y)$ both negative?"
In that case the answer is $\displaystyle \pi \left( {2K + 1} \right) < x < \pi \left( {\frac{3} {2} + 2K} \right)\;\& \;\pi \left( {2J + 1} \right) < y < \pi \left( {\frac{3} {2} + 2J} \right)$
for any two integers $\displaystyle K\;\&\;J$