sin u=1/3 and sin v=1/4.
Then cos u = sqrt(8)/3
cos v = sqrt(15)/4
sin(u+v) = ?
Trig identity:
sin(u+v)= sin u cos v + cos u sin v
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 ) ]
ty!
To find sin(u+v), we can use the trigonometric identity:
sin(u+v) = sin(u) * cos(v) + cos(u) * sin(v)
We are given that sin(u) = 1/3 and sin(v) = 1/4. We need to find cos(u) and cos(v) to substitute into the formula.
To find cos(u), we can use the Pythagorean identity:
sin^2(u) + cos^2(u) = 1
Substituting sin(u) = 1/3 into the equation, we have:
(1/3)^2 + cos^2(u) = 1
1/9 + cos^2(u) = 1
cos^2(u) = 1 - 1/9 = 8/9
Taking the square root of both sides, we get:
cos(u) = sqrt(8)/3 (positive value since cos(u) is positive when sin(u) is positive)
Similarly, to find cos(v), we use the Pythagorean identity:
sin^2(v) + cos^2(v) = 1
Substituting sin(v) = 1/4 into the equation, we have:
(1/4)^2 + cos^2(v) = 1
1/16 + cos^2(v) = 1
cos^2(v) = 1 - 1/16 = 15/16
Taking the square root of both sides, we get:
cos(v) = sqrt(15)/4 (positive value since cos(v) is positive when sin(v) is positive)
Now we can substitute the values of sin(u), sin(v), cos(u), and cos(v) into the formula for sin(u+v):
sin(u+v) = sin(u) * cos(v) + cos(u) * sin(v)
= (1/3 * sqrt(15)/4) + (sqrt(8)/3 * 1/4)
= sqrt(15)/12 + sqrt(8)/12
= (sqrt(15) + sqrt(8))/12
Therefore, sin(u+v) = (sqrt(15) + sqrt(8))/12.