SOLUTION: If tan α = −5/12 and cot β = 8/15 for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β)

Algebra ->  Trigonometry-basics -> SOLUTION: If tan α = −5/12 and cot β = 8/15 for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β)      Log On


   

Question 733572: If tan α = −5/12 and cot β = 8/15
for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β)
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
If tan α = −5/12 and cot β = 8/15
for a second-quadrant angle α and a third-quadrant angle β, find sin(α + β)
***
Identity: sin(α + β)=sin α cos β+cos α sin β
let O=opposite side
let A=adjacent side
let H=hypotenuse
..
tan α = −5/12 (in Q2)=O/-A
O=5, A=-12
H=√(O^2+A^2)=√(25+144)=√169=13
sin α=O/H=5/13
cos α=A/H=-12/13
..
cot β = 8/15(in Q3)=-A/-O
A=-8, O=-15
H=√(O^2+A^2)=√(225+64)=√289=17
sin β=O/H=-15/17
cos β=A/H=-8/17
..
sin(α + β)=sin α cos β+cos α sin β
=(5/13*-8/17)+(-12/13*-15/17)
=-40/221+180/221
sin(α + β)=140/221
..
Check with calculator:
tan α=-5/12 (in Q2)
α≈157.38º
cot β = 8/15(in Q3)
β≈241.93º
α+β≈157.38+241.93≈399.31
reference angle in Q1≈39.31º
sin 39.31=0.6335
sin(α + β)=140/221≈0.6335