If  and , what would the exact value of  be?


The tangent is positive (since it's given as +2) and the sine is negative,
(since we are given . The tangent is positive in the
1st and 3rd quadrants, and the sine is negative in the 3rd and 4th, so
 must be in the 3rd quadrant, or Q3.  So we draw an angle in
the 3rd quadrant like below.  Since the tangent is the opposite over the
adjacent, or , we make the given tangent, 2, into a fraction , but 
x goes left and y goes down, so they both have to be negative, so we
change the  to  and make the numerator -2 be the value of y
and the denominator -1 be the value of x, so we have:    
 


Now we just need to know that the cosecant is the hypotenuse over the
opposite or r/y, which is  or . 

 = 
 
Edwin