# trig

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Prove that if Sinx=5/13 and Siny=12/13, where x and y angles in Q1, then Sin(x+y)=1

• trig -

sin ( x ) = 5 / 12

cos ( x ) = + OR - sqrt [ 1 - sin ^ 2 ( x ) ]

cos ( x ) = + OR - sqrt [ 1 - ( 5 / 1 3 ) ^ 2 ]

cos ( x ) = + OR - sqrt ( 1 - 25 / 169 )

cos ( x ) = + OR - sqrt ( 169 / 169 - 25 / 169 )

cos ( x ) = + OR - sqrt ( 144 / 169 )

cos ( x ) = + OR - 12 / 13

In quaqdrant I cosine are positive so

cos ( x ) = 12 / 13

sin ( y ) = 12 / 13

cos ( y ) = + OR - sqrt [ 1 - sin ^ 2 ( y ) ]

cos ( y ) = + OR - sqrt [ 1 - ( 12 / 1 3 ) ^ 2 ]

cos ( y ) = + OR - sqrt ( 1 - 144 / 169 )

cos ( y ) = + OR - sqrt ( 169 / 169 - 144 / 169 )

cos ( y ) = + OR - sqrt ( 25 / 169 )

cos ( y ) = + OR - 5 / 13

In quaqdrant I cosine are positive so

cos ( y ) = 5 / 13

sin ( x + y ) = sin ( x ) * cos ( y ) + cos ( x ) * sin ( y )

sin ( x + y ) = ( 5 / 13 ) * ( 5 / 13 ) + ( 12 / 13 ) * ( 12 / 13 )

sin ( x + y ) = 25 / 169 + 144 / 169

sin ( x + y ) = 169 / 169 = 1

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