Question 1154735

the exact value of 

{{{sin(x-y)}}} if {{{sin(x) = 4/9}}} and {{{sin(y) = 1/4}}}

use identities:

{{{sin(x-y)=sin(x) cos(y) - cos(x) sin(y)}}}
and
{{{cos^2(x)=1-sin^2(x)}}}


if {{{sin(x) = 4/9}}}, then {{{cos^2(x)=1-(4/9)^2=1-16/81=65/81}}}

=> {{{cos(x)=sqrt(65/81)=sqrt(65)/9}}}

{{{sin(y) = 1/4}}}

{{{cos^2(y)=1-(1/4)^2}}}

{{{cos^2(y)=1-1/16}}}

{{{cos^2(y)=15/16}}}

{{{cos(y)=sqrt(15/16)}}}

{{{cos(y)=sqrt(15)/4}}}

Therefore,

{{{sin(x-y)=(4/9) (sqrt(15)/4) - (1/4)(sqrt(65)/9) }}}

{{{sin(x-y)= sqrt(15)/9 - sqrt(65)/36 }}}

{{{sin(x-y)= 4sqrt(15)/36 - sqrt(65)/36 }}}

{{{sin(x-y)= (4sqrt(15) - sqrt(65))/36 }}}-> the exact value