Find the exact value of cos(u+v) given that sine u=4/5 with u in quadrant II and sine v = -12/13 with v in quadrant IV.
Not sure how to even begin.
sin u = 4 / 5
cos u = sqroot ( 1 - sin ^ 2 u )
cos u = sqroot [ 1 - ( 4 / 5 ) ^ 2 ]
cos u = sqroot ( 1 - 16 / 25 )
cos u = sqroot ( 25 / 25 - 16 / 25 )
cos u = sqroot ( 9 / 25 )
cos u = ± 3 / 5
In Quadrant II cos is negative so:
cos u = - 3 / 5
sin v = - 12 /13
cos v = sqroot ( 1 - sin ^ 2 v )
cos v = sqroot [ 1 - ( - 12 / 13 ) ^ 2 ]
cos v = sqroot ( 1 - 144 / 169 )
cos v = sqroot ( 169 / 169 - 144 / 169 )
cos v = sqroot ( 25 / 169 )
cos v = ± 5 / 13
In Quadrant IV cos is postive so:
cos v = 5 / 13
sin u = 4 / 5, cos u = - 3 / 5, sin v = - 12 /13, cos v = 5 / 13,
cos ( u + v ) = cos u * cos v - sin u * sin v
cos ( u + v ) = ( - 3 / 5 ) * ( 5 / 13 ) - ( 4 / 5 ) * ( - 12 / 13 ) =
- 3 / 13 + 48 / 65 =
- 3 * 5 / ( 13 * 5 ) + 48 / 65 =
- 15 / 65 + 48 / 65 = 33 / 65
cos ( u + v ) = 33 / 65
To find the exact value of cos(u+v), we can use the identities involving cosine and sine. In particular, we have the following identity:
cos(u+v) = cos(u)*cos(v) - sin(u)*sin(v)
Now, let's find the values of cos(u) and cos(v) using the given information:
Since sin(u) = 4/5, we can use the Pythagorean identity to find cos(u):
cos^2(u) = 1 - sin^2(u)
cos^2(u) = 1 - (4/5)^2
cos^2(u) = 1 - 16/25
cos^2(u) = 9/25
Taking the positive square root:
cos(u) = √(9/25)
cos(u) = 3/5
Similarly, since sin(v) = -12/13, we can find cos(v):
cos^2(v) = 1 - sin^2(v)
cos^2(v) = 1 - (-12/13)^2
cos^2(v) = 1 - 144/169
cos^2(v) = 25/169
Taking the positive square root:
cos(v) = √(25/169)
cos(v) = 5/13
Now, substitute these values back into the identity:
cos(u+v) = cos(u)*cos(v) - sin(u)*sin(v)
cos(u+v) = (3/5)*(5/13) - (4/5)*(-12/13)
Simplifying the expression:
cos(u+v) = 3/13 + 48/65
cos(u+v) = (3*5 + 48)/65
cos(u+v) = 15/65 + 48/65
cos(u+v) = 63/65
Therefore, the exact value of cos(u+v) is 63/65.
To find the exact value of cos(u+v), we can use the trigonometric identity:
cos(u+v) = cos(u)cos(v) - sin(u)sin(v)
First, we need to find the values of cos(u) and cos(v).
Since sine u = 4/5, we can use the Pythagorean identity to find the value of cos(u):
cos^2(u) + sin^2(u) = 1
cos^2(u) + (4/5)^2 = 1
cos^2(u) + 16/25 = 1
cos^2(u) = 1 - 16/25
cos^2(u) = 25/25 - 16/25
cos^2(u) = 9/25
Since u is in quadrant II (where cosine is negative), we can conclude that cos(u) = -3/5.
Similarly, using the same method, we can find the value of cos(v):
cos^2(v) = 1 - sin^2(v)
cos^2(v) = 1 - (-12/13)^2
cos^2(v) = 1 - 144/169
cos^2(v) = 25/169
Since v is in quadrant IV (where cosine is positive), we can conclude that cos(v) = √(25/169) = 5/13.
Now, substituting the values of cos(u), cos(v), sin(u), and sin(v) into the trigonometric identity, we have:
cos(u+v) = cos(u)cos(v) - sin(u)sin(v)
cos(u+v) = (-3/5)(5/13) - (4/5)(-12/13)
Simplifying the expression:
cos(u+v) = -15/65 + 48/65
cos(u+v) = 33/65
Therefore, the exact value of cos(u+v) is 33/65.
first, draw your triangles in standard position. Then it is easy to see that
cos(u) = -3/5
cos(v) = 5/13
Then recall your sum formula:
cos(u+v) = cosu cosv - sinu sinv
now just plug in your numbers.