Question 1159221
If tan α = − 4/3 and cot β = 15/8
 for a second-quadrant angle α and a third-quadrant angle β, find the following.
1. sin (a+b)
2. cos (a+b)
3. tan (a+b)

I got -5 for #1, 20 for #2, and 13/84 for #3. They are all wrong and I do not know why. Maybe because I just substituted the fractions I was given into the equations?
<pre>           {{{matrix(1,7, tan (alpha), "=", O/A, "=", 4/(- 3), "=", y/x)}}}
           As this is a 3-4-5 right-triangle, r = 5. This triangle exists in the 2nd quadrant 
Therefore, {{{matrix(2,7, cos (alpha), "=", A/H, "=", x/r, "=", (- 3)/5,
sin (alpha), "=", O/H, "=", y/r, "=", 4/5)}}} 

           {{{matrix(1,7, cot (beta), "=", A/O, "=", (- 15)/(- 8), "=", x/y)}}}
           As this is an 8-15-17 right-triangle, r = 17. This triangle exists in the 3rd quadrant, where cos and sin are < 0. 
Therefore, {{{matrix(3,7, cos (beta), "=", A/H, "=", x/r, "=", (- 15)/17,
sin (beta), "=", O/H, "=", y/r, "=", (- 8)/17, tan (beta), "=", O/A, "=", y/x, "=", 8/15)}}}
 
<b>1.</b>  {{{matrix(1,3, sin (alpha + beta), "=", sin (alpha) cos (beta) + cos (alpha) sin (beta))}}} 
    {{{matrix(1,10, highlight(sin (alpha + beta)), "=", (4/5) * ((- 15)/17) + ((- 3)/5) * ((- 8)/17), "=", (- 60 + 24)/5(17), "=", - 36/5(17), ",", or, highlight(- 36/85)))}}} 
<b>2.</b>  {{{matrix(1,3, cos (alpha + beta), "=", cos (alpha) cos (beta) - sin (alpha) sin (beta))}}} 
    {{{matrix(1,10, highlight(cos (alpha + beta)), "=", ((- 3)/5) * ((- 15)/17) - (4/5) * ((- 8)/17), "=", (45 - - 32)/5(17), "=", 77/5(17), ",", or, highlight(77/85))}}}
<b>3.</b>  {{{matrix(1,3, tan (alpha + beta), "=", (tan (alpha) + tan (beta))/(1 - tan (alpha) tan (beta)))}}} 
    {{{highlight_green(matrix(1,17, highlight(tan (alpha + beta)), "=", ((- 4/3) + (8/15))/(1 - (- 4/3) * (8/15)), "=", ((- 4(5) + 8)/15)/(1 - (- 32/45)), "=", (- 12/15)/(1 + 32/45), "=", (- 4/5)/(45/45 + 32/45), "=", (- 4/5)/(77/45), "=", (- 4/5) * (45/77), "=", (- 4/cross(5)) * (9cross(45)/77), "=", highlight(- 36/77)))}}}