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If sinx +siny=a and cosx +cosy=b then find sin(x+y)and cos(x+y)

If sinx +siny=a and cosx +cosy=b then find sin(x+y)and cos(x+y)

Grade:11

1 Answers

Arun
25763 Points
4 years ago
sinx + siny = a      
 
2sin(x+y/2)cos(x-y/2) = a            ......................1                   (by using sum formula)
 
 
 
cosx + cosy = b
 
2cos(x-y/2)cos(x+y/2) = b                    ..................2
 
 
 
divide both
 
tan(x+y/2) = a/b               ...........3
 
tan2@ = sec2@ - 1           &         sec@ = 1/cos@ so
 
cos(x+y/2) = [b2/a2+b2]1/2 ......................4
 
sin(x+y/2) = {1-cos2(x+y/2)}1/2 = [a2/a2+b2]1/2.............5
 
 
 
sinx+y = 2sin(x+y/2)cos(x+y/2)                                    (sinA = 2sinA/2cosA/2)
 
from eq 4 & 5
 
sin(x+y )= 2ab/a2+b2
 
similarily we can deduce cos (x +y)

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