Question 884510
<pre>
{{{sin(arctan(5/sqrt(15)) - arcsin(3/sqrt(10)))) }}}

Let {{{alpha}}}{{{""=""}}}{{{arctan(5/sqrt(15))}}}

and {{{beta}}}{{{""=""}}}{{{arcsin(3/sqrt(10))}}}

{{{arctan(5/sqrt(15))}}} means the angle whose tangent is {{{5/sqrt(15)}}}.
The tangent is {{{opposite/(adjacent)}}} so let's draw a right triangle
containing angle {{{alpha}}} by using the numerator of {{{5/sqrt(15)}}}, which
is 5, for the opposite side, and using the denominator of {{{5/sqrt(15)}}},
which is {{{sqrt(15)}}} for the opposite side.

{{{arcsin(3/sqrt(10))}}} means the angle whose sine is {{{3/sqrt(10)}}}.
The sine is {{{opposite/(hypotenuse)}}} so let's draw another right triangle
containing angle {{{beta}}} by using the numerator of {{{3/sqrt(10)}}}, which is
3, for the opposite side, and using the denominator of {{{3/sqrt(10)}}}, which
is {{{sqrt(10)}}} for the hypotenuse.


{{{drawing(1200/7,200,-1,5,-1,6,
locate(1,.8,alpha), locate(4,2.6,5),locate(2,-.1,sqrt(15)), 
triangle(0,0,sqrt(15),0, sqrt(15), 5))}}}{{{drawing(1200/7,200,-1,5,-1,6,
locate(.3,.7,beta), locate(1.1,1.5,3),locate(-.4,2,sqrt(10)), 
triangle(0,0,1,0,1, 3))}}}


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

{{{c^2=a^2+b^2}}}
{{{c^2=(sqrt(15))^2+5^2}}}
{{{c^2=15+25}}}
{{{c^2=40}}}
{{{c=sqrt(40)}}}
{{{c=sqrt(4*10)}}}
{{{c=2sqrt(10)}}}

{{{c^2=a^2+b^2}}}
{{{(sqrt(10))^2=a^2+3^2}}}
{{{10=a^2+9}}}
{{{1=a^2}}}
{{{sqrt(1)=a}}}
{{{1=a}}}

Put those values on the triangles:

{{{drawing(1200/7,200,-1,5,-1,6,locate(.5,2.6,2sqrt(10)),
locate(1,.8,alpha), locate(4,2.6,5),locate(2,-.1,sqrt(15)), 
triangle(0,0,sqrt(15),0, sqrt(15), 5))}}}{{{drawing(1200/7,200,-1,5,-1,6,
locate(.3,.7,beta), locate(1.1,1.5,3),locate(-.4,2,sqrt(10)), 
triangle(0,0,1,0,1, 3), locate(.5,0,1))}}}

Now

{{{sin(arctan(5/sqrt(15)) - arcsin(3/sqrt(10)))) }}}{{{""=""}}}{{{sin(alpha-beta)}}}{{{""=""}}}{{{sin(alpha)cos(beta)-cos(alpha)sin(beta)}}}{{{""=""}}}

{{{( 5/(2sqrt(10)) )(1/sqrt(10))-(sqrt(15)/(2sqrt(10)))(3/sqrt(10))}}} {{{""=""}}} {{{5/(2*10)}}}{{{""-""}}}{{{3sqrt(15)/(2*10)}}} {{{""=""}}} {{{5/20}}}{{{""-""}}}{{{3sqrt(15)/20}}} {{{""=""}}} {{{(5-3sqrt(15))/20}}}

Edwin</pre>