Question 1204118



{{{tan(x) = -40/9}}}

{{{tan(x) = sin(x)/cos(x)}}}

also, 

{{{tan(x) = opp/adj}}}

{{{sin (x)= opp / hyp }}}

{{{cos (x) = adj / hyp}}}

then {{{opp/adj= -40/9 }}}=> {{{opp=-40}}}, {{{adj=9}}}

=> {{{hyp=sqrt((-40)^2+9^2)=41}}}


then 

{{{sin (x)= -40/ 41}}}

{{{cos (x) = 9 / 41 }}}


(a)

{{{sin(x/2)}}}

using Half Angle Formulas

{{{sin (x/2) =sqrt ( (1 - cos(x)) / 2 )}}}

{{{sin (x/2) = ( (1 -(9 / 41)) / 2 )}}}

{{{sin (x/2) = sqrt ( 16/41 )}}}

{{{sin (x/2) =  4/sqrt ( 41 )}}} or {{{sin (x/2) =  - 4/sqrt ( 41 )}}}

sin is negative in Q IV

{{{sin (x/2) = - 4/sqrt ( 41 )}}}



(b)

{{{cos(x/2)}}}

{{{cos (x/2) = sqrt ( (1 + (9 / 41)) / 2 )}}}
 
{{{cos (x/2) = sqrt ( (50/41) / 2 ) }}}

{{{cos (x/2) = sqrt ( 25/41 ) }}}

{{{cos (x/2) = 5/sqrt(41)}}}...cos is positive in Q IV



(c)

{{{tan(x/2)=-(4/sqrt ( 41 ))/(5/ sqrt(41))=-4/5}}}