SOLUTION: Find the exact value of the following under the given conditions. if tan(x)= -4/3 and (pi/2)< x < pi and if sin(y)= 2/3 and 0< y < (pi/2) find tan(x+y) show wor

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of the following under the given conditions. if tan(x)= -4/3 and (pi/2)< x < pi and if sin(y)= 2/3 and 0< y < (pi/2) find tan(x+y) show wor      Log On


   

Question 965366: Find the exact value of the following under the given conditions.
if tan(x)= -4/3
and (pi/2)< x < pi
and
if sin(y)= 2/3
and 0< y < (pi/2)
find tan(x+y)
show work and state exact value
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Find the exact value of the following under the given conditions.
if tan(x)= -4/3
and (pi/2)< x < pi
and
if sin(y)= 2/3
and 0< y < (pi/2)
find tan(x+y)
***
tanx=-4/3 ((3-4-5) reference right triangle in quadrant II)
sinx=4/5
cosx=-3/5
..
siny=2/3
cosy=sqrt%281-sin%5E2%28y%29%29=sqrt%281-4%2F9%29=sqrt%285%2F9%29=sqrt%285%29%2F3
..
sin(x+y)=sinx*cosy+cosx*siny=4/5*√5/3+-3/5*2/3=4√5/15-6/15=(4√5-6)/15
cos(x+y)=cosx*cosy-sinx*siny=-3/5*√5/3-4/5*2/3=-3√5/15-8/15=-(3√5+8)/15
tan(x+y)=sin(x+y)/cos(x+y)=-(4√5-6)/(3√5+8)
..
check:
tanx=-4/3 (Q2)
x=126.87˚
siny=2/3 (Q1)
y=41.81˚
x+y=168.68
tan(x+y)=tan(168.68)≈-0.2002
exact value as computed above=-(4√5-6)/(3√5+8)≈-0.2002