Question 958899
<pre>
sin[tan<sup>-1</sup>(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.

{{{drawing(400,200,-1,9,-1,4,
red(arc(0,0,4,-4,0,21)),rectangle(7.7,0,8,.3),
triangle(0,0,8,0,8,3), locate(8.1,1.5,3), locate(4,0,8)  )}}}
 
The angle marked with the red arc is tan<sup>-1</sup>(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:

{{{hypotenuse^2=adjacent^2+opposite^2}}}

{{{hypotenuse^2=8^2+3^2}}}

{{{hypotenuse^2=64+9}}}

{{{hypotenuse^2=73}}}

{{{hypotenuse=sqrt(73)}}}

So now we can label the hypotenuse:

{{{drawing(400,200,-1,9,-1,4, locate(4,2.25,sqrt(73)),
red(arc(0,0,4,-4,0,21)),rectangle(7.7,0,8,.3),
triangle(0,0,8,0,8,3), locate(8.1,1.5,3), locate(4,0,8)  )}}}

Now since the original problem is

sin[tan<sup>-1</sup>(3/8)]

We find the sine by putting the opposite side 3 over the hypotenuse &#8730;<span style="text-decoration: overline">73</span>

So the answer is 

sin[tan<sup>-1</sup>(3/8)] = 3/&#8730;<span style="text-decoration: overline">73</span>

If you like you can rationalize the denominator and get 3&#8730;<span style="text-decoration: overline">73</span>/73

Edwin</pre>