I will 'define' the trigonometric ratios in this way.
Suppose that I am standing and holding a rod of length $1$ unit in front of me, with one end of the rod next to my foot on the ground. The rod has an inclination of $\theta$ to the ground. $\sin\theta$ is the distance between the other end of the rod and the ground. The sun is directly above me and casting a shadow of the rod on the ground. $\cos\theta$ is the length of the shadow. $\tan\theta$ is the quotient of the two lengths.
If $\theta$ is obtuse, the rod is behind me, with one end still next to my foot. The distance between the other end of the rod and the ground is still $\sin\theta$, and it is positive. The shadow of the rod is now behind me, and I take its length as negative. This is $\cos\theta$ and we have $\cos\theta<0$. $\tan\theta$ is the quotient of this two lengths, which is also negative.
I can extend this 'definition' to a general $\theta$. If I hold the rod vertically, $\theta=90^\circ$. $\sin90^\circ$ is the distance between the other end of the rod and the ground, which is now equal to the length of the rod. So $\sin90^\circ =1$. The length of the shadow is $0$ and so $\cos90^\circ =0$.
If we take $\theta=200^\circ$, the rod is below the ground (we have to imagine that). So $\sin200^\circ$ is negative....