Question 734106: Cos(arcsin(5/13)-arctan(3/4))
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Cos(arcsin(5/13)-arctan(3/4))
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let O=opposite side
let A=adjacent side
let H=hypotenuse
let angle x=arcsin(5/13)
let angle y=arctan(3/4)
..
sin x=5/13=O/H
O=5, H=13
A=√(h^2-O^2)=√(13^2-5^2)=√(169-25)=√144=12
cos x=A/H=12/13
..
tan y=3/4=O/A
O=3, A=4
H=(O^2+A^2)=√(3^2+4^2)=√9+16)=√25=5
cos y=A/H=4/5
sin y=O/H=3/5
..
Identity: cos (x-y)=(cos x cos y)+(sin x sin y)
=(12/13*4/5)+(5/13*3/5)
=48/65+15/65
=63/65
..
Check with calculator:
arcsin(5/13)≈22.62º
arctan(3/4)≈36.87º
(arcsin(5/13)-arctan(3/4))≈-14.25º
cos (-14.25)º≈0.9692
63/65≈0.9692
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