Question 1161413
If {{{cos (x)= -15/17 }}}with {{{x }}}in quadrant III, find the exact value of {{{sin (2x)}}} and the quadrant of {{{2x}}}

given:

{{{cos (x)= -15/17 }}}

find:

{{{sin (2x)}}} 

use identity:

{{{sin(2x)=2sin(x)cos(x)}}} ......substitute cos

{{{sin(2x)=2sin(x)(-15/17) }}}


 use  {{{sin(x)=sqrt(1-cos^2(x))}}} to find sin

{{{sin(x)=sqrt(1-(-15/17) ^2)}}}

{{{sin(x)=sqrt(1-225/289)}}}

{{{sin(x)=sqrt((289-225)/289)}}}

{{{sin(x)=sqrt(64/289)}}}

{{{sin(x)}}}=±{{{8/17)}}}


In Quadrant III, sine and cosine are negative:

{{{sin(x)=-8/17}}}

then

{{{sin(2x)=2(-8/17)(-15/17) }}}

{{{sin(2x)=240/289}}}

{{{2x=sin^-1(240/289)}}}

{{{2x=56.14}}}° -> In Quadrant {{{I}}}