SOLUTION: Evaluate exactly: sin[arctan(5/sqrt15) - arcsin(3/sqrt10)]

Let 

and 

 means the angle whose tangent is .
The tangent is  so let's draw a right triangle
containing angle  by using the numerator of , which
is 5, for the opposite side, and using the denominator of ,
which is  for the opposite side.

 means the angle whose sine is .
The sine is  so let's draw another right triangle
containing angle  by using the numerator of , which is
3, for the opposite side, and using the denominator of , which
is  for the hypotenuse.





We calculate the hypotenuse of the first right triangle and
the adjacent side to  in the second one:
















Put those values on the triangles:



Now



      

Edwin