Question 963649
<pre>
A little more detail.  It's good to draw triangles
on x and y axes for such problems:

sin(arccos(3/5) - arctan(5/12))

Let A = arccos(3/5) and B = arctan(5/12)

The we use the identity:

sin(A-B)= sin(A)cos(B)-cos(A)sin(B)

First we draw A = arccos(3/5).

arccos(3/5) means 

"the angle in the first quadrant whose cosine is 3/5".

So we draw a triangle in the first quadrant.  Since 
cosine = adj/hyp = x/r we make the adjacent side, x, the
same as the numerator of 3/5, which is x=3 and make 
the hypotenuse, r, the denominator of 3/5 which is r=5.

{{{drawing(1000/3,400,-1,4,-1,5,
line(-5,0,7,0), line(0,-3,0,7),line(0,0,3,4),line(3,4,3,0),
red(arc(0,0,2,-2,0,54)), locate(1.5,0,x=3), locate(3.1,2,"y=?"),
locate(.8,2,r=5)


 )}}}{{{


matrix(14,1,

By,the,Pythagorean, theorem,we,calculate,
y, by, x^2+y^2=r^2,
3^2+y^2=5^2, 9+y^2=25,
y^2=16,y=sqrt(16),y=4) }}}{{{drawing(1000/3,400,-1,4,-1,5,
line(-5,0,7,0), line(0,-3,0,7),line(0,0,3,4),line(3,4,3,0),
red(arc(0,0,2,-2,0,54)), locate(1.5,0,x=3), locate(3.1,2,y=4),
locate(.8,2,r=5)


 )}}}
 

Next we draw arctan(5/12).

arctan(5/12) means 

"the angle in the first quadrant whose tangent is 5/12".

So we draw another right triangle in the first quadrant.  
Since tangent = opp/adj = y/x we make the opposite side, y, the
same as the numerator of 5/12, which is y=5 and make 
the adjacent side, x, the denominator of 5/12 which is x=12.


{{{drawing(400,200,-1,13,-1,6,
line(-5,0,15,0), line(0,-3,0,15),line(0,0,12,5),line(12,5,12,0),
red(arc(0,0,5,-5,0,23)), locate(6,0,x=12), locate(12.1,2.5,y=5),
locate(6,3.5,"r=?")


 )}}}{{{


matrix(14,1,

By,the,Pythagorean, theorem,we,calculate,
r, by, x^2+y^2=r^2,
5^2+12^2=r^2, 25+144=r^2,
169=r^2,sqrt(169)=r,13=r) }}}{{{drawing(400,200,-1,13,-1,6,
line(-5,0,15,0), line(0,-3,0,15),line(0,0,12,5),line(12,5,12,0),
red(arc(0,0,5,-5,0,23)), locate(6,0,x=12), locate(12.1,2.5,y=5),
locate(6,3.5,r=13)


 )}}} 

Now we can easily finish:

sin(A-B)= sin(A)cos(B)-cos(A)sin(B)
        = (4/5)(12/13)-(3/5)(5/13)
        = 48/65 - 15/65
        = 33/65


Edwin</pre>