Trig identity:
sin(u+v)= sin u cos v + cos u sin v
sin u=1/3 and sin v=1/4.
Then cos u = sqrt(8)/3
cos v = sqrt(15)/4
sin(u+v) = ?
4 answers
i did that and still have the wrong answer, don't know why
sin u = 1 / 3
cos u = sqroot [ 1 - sin u ) ^ 2 ]
sqroot [ 1 - ( 1 / 3 ) ^ 2 ] =
sqroot ( 1 - 1 / 9 ) =
sqroot ( 9 / 9 - 1 / 9 ) =
sqroot ( 8 / 9 ) =
sqroot ( 4 * 2 / 9 ) =
sqroot ( 4 / 9 ) * sqroot ( 2 ) =
+ OR - ( 2 / 3 ) * sqroot ( 2 )
cos u = + OR - 2 sqrt ( 2 ) / 3
sin v = 1 / 4
cos v = sqroot [ 1- ( sin v ) ^ 2 ]
sqroot [ 1 - ( 1 / 4 ) ^ 2 ] =
sqroot ( 1 - 1 / 16 ) =
sqroot ( 16 / 9 - 1 / 16 ) =
sqroot ( 15 / 16 ) =
+ OR - sqroot ( 15 ) / 4
cos v = + OR - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
You have 5 solutions :
1.
sin u is positive, cos u is positive, sin v is positive, cos v is positive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = 1 / 4, cos v = sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * sqroot ( 15 ) / 4 + 2 sqrt ( 2 ) / 3 * ( 1 / 4 ) =
( 1 / 12 ) * sqroot ( 15 ) + 2 sqrt ( 2 ) / 12
sin ( u + v ) = ( 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
2.
sin u is positive, cos u is positive, sin v is positive, cos v is negaitive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] + 2 sqrt ( 2 ) / 12
sin ( u + v ) = ( 1 / 12 ) [ 2 sqroot ( 2 ) - sqroot ( 15 ) ]
3.
sin u is positive, cos u is positive, sin v is negative, cos v is negaitive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] - 2 sqrt ( 2 ) / 12 =
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
4.
sin u is positive, cos u is negative, sin v is negative, cos v is negaitive:
sin u = 1 / 3, cos u = - 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] + 2 sqrt ( 2 ) / - 12 =
( - 1 / 12 ) * [ sqroot ( 15 ) ] + 2 sqrt ( 2 ) / - 12 =
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqrt ( 2 ) ]
5.
sin u is negaitive, cos u is negative, sin v is negative, cos v is negaitive:
sin u = - 1 / 3, cos u = - 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( - 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
[ - sqroot ( 15 ) / - 3 *4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
sqroot ( 15 ) / 12 + 2 sqrt ( 2 ) / -12 =
sin ( u + v ) = ( 1 / 12 ) * [ sqroot ( 15 ) - 2 sqrt ( 2 ) ]
Solutions 3 and 4 are same solution.
This mean you have total 4 solutions:
sin ( u + v ) = ( 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
sin ( u + v ) = ( 1 / 12 ) [ 2 sqroot ( 2 ) - sqroot ( 15 ) ]
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqrt ( 2 ) ]
sin ( u + v ) = ( 1 / 12 ) * [ sqroot ( 15 ) - 2 sqrt ( 2 ) ]
cos u = sqroot [ 1 - sin u ) ^ 2 ]
sqroot [ 1 - ( 1 / 3 ) ^ 2 ] =
sqroot ( 1 - 1 / 9 ) =
sqroot ( 9 / 9 - 1 / 9 ) =
sqroot ( 8 / 9 ) =
sqroot ( 4 * 2 / 9 ) =
sqroot ( 4 / 9 ) * sqroot ( 2 ) =
+ OR - ( 2 / 3 ) * sqroot ( 2 )
cos u = + OR - 2 sqrt ( 2 ) / 3
sin v = 1 / 4
cos v = sqroot [ 1- ( sin v ) ^ 2 ]
sqroot [ 1 - ( 1 / 4 ) ^ 2 ] =
sqroot ( 1 - 1 / 16 ) =
sqroot ( 16 / 9 - 1 / 16 ) =
sqroot ( 15 / 16 ) =
+ OR - sqroot ( 15 ) / 4
cos v = + OR - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
You have 5 solutions :
1.
sin u is positive, cos u is positive, sin v is positive, cos v is positive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = 1 / 4, cos v = sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * sqroot ( 15 ) / 4 + 2 sqrt ( 2 ) / 3 * ( 1 / 4 ) =
( 1 / 12 ) * sqroot ( 15 ) + 2 sqrt ( 2 ) / 12
sin ( u + v ) = ( 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
2.
sin u is positive, cos u is positive, sin v is positive, cos v is negaitive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] + 2 sqrt ( 2 ) / 12
sin ( u + v ) = ( 1 / 12 ) [ 2 sqroot ( 2 ) - sqroot ( 15 ) ]
3.
sin u is positive, cos u is positive, sin v is negative, cos v is negaitive:
sin u = 1 / 3, cos u = 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] - 2 sqrt ( 2 ) / 12 =
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
4.
sin u is positive, cos u is negative, sin v is negative, cos v is negaitive:
sin u = 1 / 3, cos u = - 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
( 1 / 12 ) * [ - sqroot ( 15 ) ] + 2 sqrt ( 2 ) / - 12 =
( - 1 / 12 ) * [ sqroot ( 15 ) ] + 2 sqrt ( 2 ) / - 12 =
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqrt ( 2 ) ]
5.
sin u is negaitive, cos u is negative, sin v is negative, cos v is negaitive:
sin u = - 1 / 3, cos u = - 2 sqrt ( 2 ) / 3, sin v = - 1 / 4, cos v = - sqroot ( 15 ) / 4
sin ( u + v ) = sin u * cos v + cos u * sin v
sin ( u + v ) = ( - 1 / 3 ) * [ - sqroot ( 15 ) / 4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
[ - sqroot ( 15 ) / - 3 *4 ] + 2 sqrt ( 2 ) / 3 * ( - 1 / 4 ) =
sqroot ( 15 ) / 12 + 2 sqrt ( 2 ) / -12 =
sin ( u + v ) = ( 1 / 12 ) * [ sqroot ( 15 ) - 2 sqrt ( 2 ) ]
Solutions 3 and 4 are same solution.
This mean you have total 4 solutions:
sin ( u + v ) = ( 1 / 12 ) [ sqroot ( 15 ) + 2 sqroot ( 2 ) ]
sin ( u + v ) = ( 1 / 12 ) [ 2 sqroot ( 2 ) - sqroot ( 15 ) ]
sin ( u + v ) = ( - 1 / 12 ) [ sqroot ( 15 ) + 2 sqrt ( 2 ) ]
sin ( u + v ) = ( 1 / 12 ) * [ sqroot ( 15 ) - 2 sqrt ( 2 ) ]
ty!