SOLUTION: if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?

Algebra ->  Trigonometry-basics -> SOLUTION: if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?       Log On


   

Question 803728: if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
if x is in the first and y in the third quadrant, sinx=3/5, and cosy=-5/13, find the exact value of sin(x+y), cos(x+y), and tan(x+y)?
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sinx=3/5
so, you are working with a 3-4-5 reference right triangle in quadrant
cosx=4/5
tanx=sinx/cosx=3/4
..
cosy=-5/13
so, you are working with a 5-12-13 reference right triangle in quadrant III
siny=-12/13
tany=-12/-5=12/5
..
Identity: sin(x+y)=sinx*cosy+cosx*siny
%283%2F5%29%28-5%2F13%29%2B%284%2F5%29%28-12%2F13%29
%28-15%2F65%29%2B%28-48%2F65%29=-63%2F65
..
Identity: cos(x+y)=cosx*cosy-sinx*siny
%284%2F5%29%28-5%2F13%29-%283%2F5%29%28-12%2F13%29
%28-20%2F65%29-%28-36%2F65%29=16%2F65
..
Identity: tan(x+y)=%28tanx%2Btany%29%2F%281-tanx%2Atany%29
%28%283%2F4%29%2B%2812%2F5%29%29%2F%281-%283%2F4%29%2A%2812%2F5%29%29
%28%2815%2F20%29%2B%2848%2F20%29%29%2F%281-%2836%2F20%29%29
%2863%2F20%29%2F%28-16%2F20%29=-63%2F16
..
Calculator check:
sinx=3/5
x≈36.87˚
cosy=-5/13
y≈67.38˚+180≈247.38˚
x+y≈284.25˚
..
sin(x+y)≈sin(284.25˚)≈-0.9692..
Exact value as calculated=-63/65≈-0.9692..
..
cos(x+y)≈cos(284.25)≈0.2461..
Exact value as calculated=16/65≈0.2461..
..
tan(x+y)≈tan(284.25)≈-3.9375..
Exact value as calculated=-63/16≈-3.9375..