SOLUTION: Find the exact value of the expression sin((tan^-1 3/4) +(cos^-1 5/13))

Question 819578: Find the exact value of the expression
sin((tan^-1 3/4) +(cos^-1 5/13))
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Find the exact value of the expression
sin((tan^-1 3/4) +(cos^-1 5/13))
***
let x=angle whose tan=3/4
let y=angle whose cos=5/13
Identity: sin(x+y)=sinxcosy+cosxsiny
..
tanx=3/4(this is from a (3-4-5) right triangle)
sinx=3/5
cosx=4/5
..
cosy=5/13(this is from a (5-12-13) right triangle)
siny=12/13
..
sin((tan^-1 3/4) +(cos^-1 5/13))=sin(x+y)
sin(x+y)=3/5*5/13+4/5*12/13=15/65+48/65=63/65
..
calculator check:
tanx=3/4
x≈36.87˚
cosy=5/13
x≈67.38˚
(x+y)≈104.25
sin(x+y)≈sin(104.25)≈0.9692..
Exact value as calculated=63/65≈0.9692..

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