Question 555695
sin[cos<sup>-1</sup>({{{12/13}}})].  Give as a fraction.
<pre>
First we look at the inside only:

cos<sup>-1</sup>({{{12/13}}})  

That is the answer to this question:

<i><u>What is the smallest angle in absolute value which has {{{12/13}}}
for its cosine?</i></u>

The answer to that question is an angle in the first quadrant.
We don't know right off without a calculator what that angle is.
However we can DRAW IT in standard position, because we know that
the cosine is {{{adjacent/(hypotenuse)}}} or {{{x/r}}}, so we
draw this right triangle in the first quadrant, with the adjacent 
side equal to the numerator of the fraction, x=12, and the hypotenuse
equal to the denominator of the fraction, r=13.  So we draw this:

{{{drawing(7600/11,400,-4,15,-3,8, graph(7600/11,400,-4,15,-3,8),
locate(7,.7,x=12),locate(4.9,3,r=13), 
red(locate(2.5,1.2,cos^(-1)(12/13))),
red(arc(0,0,12,-12,0,22.61986495)),

green(line(0,0,12,0), line(12,0,12,5), line(0,0,12,5)) )}}}

The angle indicated by the red arc is the angle represented by

cos<sup>-1</sup>({{{12/13}}})

[I realize that it takes a while to get used to something that
starts with "COSINE" to represent an ANGLE, but that's what
the little "-1" does. (It's not really a -1, nor is it an exponent,
but unfortunately the mathematicians of old used that notation and 
it stuck, so we are stuck with it, but it is NOT -1 and it is not
an exponent. cos<sup>-1</sup> represents an ANGLE which has what follows it as
its cosine.)]

Now let's go back to the original problem:

sin[cos<sup>-1</sup>({{{12/13}}})]

We want the SINE of that angle indicated by the red arc and
represented by cos<sup>-1</sup>({{{12/13}}}).

The sine is {{{opposite/(hypotenuse)}}}, or {{{y/r}}}, so we will 
have to find the opposite side of that angle, which is the y value.

So we call on old man Pythagorus:

 r² = x² + y²
13² = 12² + y²
169 = 144 + y²
 25 = y²
  5 = y

So we get y = 5, so we put that over on the right of the drawing:

{{{drawing(7600/11,400,-4,15,-3,8, graph(7600/11,400,-4,15,-3,8),
locate(7,.7,x=12),locate(4.9,3,r=13), 
red(locate(2.5,1.2,cos^(-1)(12/13))),
red(arc(0,0,12,-12,0,22.61986495)),
locate(12.3,2.5,y=5),
green(line(0,0,12,0), line(12,0,12,5), line(0,0,12,5)) )}}}
 
Now we can find that sine easily as {{{opposite/"hypotenuse"}}} or {{{y/r}}}
or  {{{5/13}}}

So sin[cos<sup>-1</sup>({{{12/13}}})] = {{{5/13}}}

Edwin</pre>