Question 1199044
If {{{tan (alpha) = -12/35}}} and {{{cot (beta) = 3/4}}}

for a second-quadrant angle {{{alpha}}} and a third-quadrant angle  {{{beta}}} , find the following:

(a) {{{sin(alpha + beta)}}}
(c) {{{tan(alpha + beta)}}]


{{{tan (alpha) = -12/35 =opp/adj}}}

{{{opp=12}}}
{{{adj=35}}}

{{{hyp=sqrt(12^2+35^2)=37 }}}or{{{-37}}}


{{{sin(alpha)=12/37}}} or {{{sin(alpha)=-12/37}}}

for a second-quadrant angle  {{{alpha}}}: 

{{{sin(alpha)=12/37}}}



and {{{cot (beta) = 3/4=adj/opp}}}

{{{adj=3}}}
{{{opp=4}}}

{{{hyp=sqrt(4^2+3^2)=sqrt(25)=5}}} or{{{ -5}}}

{{{cos(beta)= 3/5}}} or {{{-3/5}}}

a third-quadrant angle  (beta): 

{{{cos(beta)=-3/5}}}


so, use
{{{sin(alpha)=12/37}}}
{{{cos(beta)=-3/5}}}


(a){{{ sin(alpha + beta)=cos(-3/5) sin(12/37) + cos(12/37) sin(-3/5)=-0.2721971}}}

(c){{{ tan(alpha + beta)= sin(alpha + beta)/cos(alpha + beta)}}}


={{{-0.2721971/(cos(12/37) cos(-3/5) - sin(12/37) sin(-3/5))}}}


={{{-0.2721971/0.9622415002904475}}}


={{{-0.2828782}}}