Why are the values of $\sin (\theta)$ and $\csc(\theta)$ positive in the second quadrant? Why is the value of $\sin (\theta)$ and $\csc(\theta)$ positive in the second quadrant?
Similarly why the value of other trigonometric ratios so in other quadrants?

You may recall this image:

Now, the unit circle definition tells us that $\cos \theta$ and $\sin\theta$ are the $x$ and $y$ coordinates of the point whose radius makes angle $\theta$ with the positive $x$-axis.
Therefore, in the first quadrant ($0<\theta<90^\circ$), both $\cos\theta$ and $\sin\theta$ are positive, and in the second quadrant ($90^\circ<\theta<180^\circ$), $\cos\theta$ is negative and $\sin\theta$ is positive, and so on for third and fourth quadrants.
Define:
$$\sec\theta=\frac{1}{\cos\theta}\text{ and }\csc\theta=\frac{1}{\sin\theta}$$
Thus, the sign of $\sec\theta$ is the same as $\cos\theta$ and the sign of $\csc\theta$ is the same as $\sin\theta$.
Define:
$$\tan\theta=\frac{\sin\theta}{\cos\theta}\text{ and }\cot\theta=\frac{\cos\theta}{\sin\theta}$$
Thus, the signs of $\tan\theta$ and $\cot\theta$ are positive, if $\sin\theta$ and $\cos\theta$ have same signs, and negative, if different.
For example, for the second quadrant, $\cos\theta$ is negative and $\sin\theta$ is positive. So, $\sec\theta$ is negative and $\csc\theta$ is positive, and , $\tan\theta$ and $\cot\theta$ are negative.
Hope this helps. Ask anything if not clear :)
