sin[tan-1(3/8)]

This says "Find the sin of the angle whose tangent is 3/8"

So we sketch a right triangle that has an angle whose
tangent is 3/8.

Since the tangent is the opposite over the adjacent, and
3/8 is 3 over 8, we the opposite side as 3, and the adjacent
side as 8.


 
The angle marked with the red arc is tan-1(3/8),
the angle whose tangent is 3/8.

Now we just need the sine of that angle marked.  (Its tangent is 3/8.)

Since the sine is the opposite over the hypotenuse, we will need
to find the hypotenuse:

We use the Pythagorean theorem:











So now we can label the hypotenuse:



Now since the original problem is

sin[tan-1(3/8)]

We find the sine by putting the opposite side 3 over the hypotenuse √73

So the answer is 

sin[tan-1(3/8)] = 3/√73

If you like you can rationalize the denominator and get 3√73/73

Edwin