Simple trigonometry: negative cos and sin

Question

I have some problem with awareness of school material. How can we say that $\cos\phi$ or $\sin\phi$ would be negative? The metric - a function that takes only positive values in $\mathbb{R}$. I know that in school about that sayng that $\phi$ -- obtuse angle. But for now it is looking little bit strange.

Answer 1

To understand that we need to recall that by definition $\cos \theta$ and $\sin \theta$ are precisely the coordinates of the point $P(x,y)$ on the unitary circle such that ray OP forms an angle $\theta$ with positive $x$ axis (usually assuming as positive the counterclockwise direction).

Since the equation for the unit circle is $x^2+y^2=1$, $\cos A$ and $\sin A$ can also assume negative values, for exampleat an angle $180°$ corresponds the point $(-1,0)$ and therefore

• $\cos (180°)=-1$
• $\sin (180°)=0$