If tan α = 12/5, π < α < 3 π/2, and sin β = 2/3, 0< β < π/2, find

We draw the graphs of the angles,  is in the third
quadrant and  is in the first:

    

Next we draw perpendiculars to the x-axis, which creates right
triangles

    

Since the tangent is opposite/adjacent or y/x, we take the opposite side of 
 to be -12 (negative because it goes downward from the x-axis), and
the adjacent to be -5 (negative because it goes left from the origin).  

Since the sine is opposite/hypotenuse or y/r, we take the opposite side of 
 to be +2 (positive because it goes upward from the x-axis), and
the hypotenuse r to be 3 (the hypotenuse r is ALWAYS taken positive).

    

Next we use the Pythagorean theorem to calculate the hypotenuse r for 
and the adjacent side x for .

r² = x² + y²            r² = x² + y²
r² = (-5)²+(-12)²       3² = x² + 2²
r² = 25 + 144            9 = x² + 4  
r² = 169                 5 = x² 
 r = 13                   = x

    

Now we can answer the first two questions by merely looking at the triangles,
and knowing sine = opposite/hypotenuse and cosine = adjacent/hypotenuse,
using x for adjacent, y for opposite and r for hypotenuse:

 A) sin α = opp/hyp = y/r = (-12)/13 = -12/13
 B) cos β = adj/hyp = x/r = /3 

To do the others we have to use identities and then we can
substitute as above.  I will do C):

 C) cos 2β = cos²β-sin²β = (/3)²-(2/3)² = 5/9-4/9=1/9

Now you can do the others by yourself, using the proper identities and
the above graphs:

 D) sin α/2
 E) cos (α+β)
 F) sin (α+β)/2 

Edwin