If i'm given the quadarnt rog function(theta) how do i find the exact value?

You just need to take the inverse tan function of the answer given.  That will give you the angle.  Once you have the angle, you can find the requested function.

The quadrant is simply a way to verify you have the correct angle, because there are multiple angles that could have the same tan function.

Inverse tangent (tan^-1) is a function on all scientific calculators.  Also pay attention if you are in radian or degree mode.

Quadrant 1
tan theta = 2
csc theta = ?

The way to approach this is to represent them in a sketch. Note that in quadrant 1 the x and y coordinate values are positive. Draw a triangle in quadrant 1 the acute angle made between the x-axis and the hypothenuse is theta.
tan theta = 2 = y-value/ x-value.
This means that y = 2 and x = 1.
So the hypothenuse = sqrt(2^2 + 1^2) = sqrt(5) = 2.24

Therefore csc theta = 1/sin theta = 2.24/2 = 1.12.



Quadrant 3

tan theta=4
sin theta=?
Draw your triangle in the third quadrant. Since x and y values in the third quadrant are negative and tan theta = 4. That means that  y = -4 and x = -1. The hypothenuse = sqrt((-4)^2 + (-1)^2) = sqrt(17) = 4.12.
sin theta = y/hypothenuse = -4/4.12 = -0.9709.