Question 881256: Find the exact value of sin (x+y) if sin x = 4/5 in Quadrant II and tan y = 12/5 in Quadrant III.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find the exact value of sin (x+y) if sin x = 4/5 in Quadrant II and tan y = 12/5 in Quadrant III.
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Identity:sin(x+y)=sinx*cosy+cosx*siny
sinx=4/5, (3-4-5 reference right triangle in quadrant II in which sin>0, cos<0.)
cosx=-3/5
..
tany=12/5 (5-12-13 reference right triangle in quadrant III in which sin<0, cos<0.)
siny=-12/13
cosy=-5/13
..
sin(x+y)=4/5*-5/13+-3/5*-12/13=-20/65+36/65=16/65
..
Calculator check:
sinx=4/5 in Q2
x≈126.87˚
tany=12/5 in Q3
y≈247.38˚
x+y ≈374.25˚
sin(x+y)=sin(374.2˚)≈0.2461…
Exact value=16/65≈0.2461…
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