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?

That's because cartesian coordinate system is defined in that way. This needs explanation, I know.
We have generalised trigonometry, from only inputs between $0$ and $90$ to any real number $\theta$, by using a unit circle (Wikipedia link) on cartesian coordinate system with its centre on origin i.e. $(0,0)$.
To do this generalisation we define $\sin (\theta)$  as $y-axis$ of the the end of the point on radius of that unit circle (Khan Academy link).
Now it is by definition that in $2nd$ and $1st$ quadrant $y-coordinate$ is positive, so clearly it follows that $sine$ function must be positive in $2nd$ and $1st$quadrant as it is $y-axis$. Now since $\sin (\theta)$ is positive, so must be $\csc(\theta)$. As we have defined it as reciprocal of $\sin (\theta)$, now reciprocal  of any positive number must be a positive number($0$ is not positive).
Now we have similarly defined $\cos(\theta)$ as the $x-coordinate$ of the the end of the point on radius of that unit circle, similarly it must be positive in $1st$ and $4rth$ quadrant.
Following image may be helpful:

It is left as an exercise for you to find why other trigonometric functions ..... you understand what I mean(right)!
