Let sin u=-8/17 and cos v= 12/13, with pie<u<3pie/2. Find the exact value of cos(u+v)
both u and v are in quadrant III
from sin u = -8/17 ----> y = -8 , r = 17
x^2 + 64 = 289 --> x = -15 in quad III
cosu = -15/17
from cos v = 12/13 ---> x = 12, r = 13
y = ± 5
if v is in I, sinv = 5/13 , cosv = 12/13
if v in in IV , sinv = -5/13, cosv = 12/13
cos(u+v) = cosu cosv - sinu sinv
if v is in I :
= (-15/17)(12/13) - (-8/17)(5/13) = -140/221
if v is in IV
= (-15/17)(12/13) - (-8/17)(-5/13) = -220/221
Well, if sin u = -8/17 and cos v = 12/13, then we can find the values of sin v and cos u using the Pythagorean identities. But since you mentioned "pie," I'm gonna go ahead and assume you meant "pi." Otherwise, I would love to have a slice of pie! 😄
So, let's find sin v using sin^2 v + cos^2 v = 1:
sin^2 v + (12/13)^2 = 1
sin^2 v + 144/169 = 1
sin^2 v = 1 - 144/169
sin^2 v = 25/169
sin v = ±√(25/169)
sin v = ±5/13
Next, let's find cos u using sin^2 u + cos^2 u = 1:
(-8/17)^2 + cos^2 u = 1
64/289 + cos^2 u = 1
cos^2 u = 1 - 64/289
cos^2 u = 225/289
cos u = ±√(225/289)
cos u = ±15/17
Now, let's find cos(u + v) using the sum formula: cos(u + v) = cos u * cos v - sin u * sin v.
cos(u + v) = (15/17) * (12/13) - (-8/17) * (5/13)
cos(u + v) = 180/221 + 40/221
cos(u + v) = 220/221
So, the exact value of cos(u + v) is 220/221. Now, that's what I call a fractionally funny answer! 🤡
To find the exact value of cos(u+v), we can start by using the sum formula for cosine:
cos(u+v) = cos u * cos v - sin u * sin v
Given that sin u = -8/17 and cos v = 12/13, we can substitute these values into the formula:
cos(u+v) = (cos u * cos v) - (sin u * sin v)
cos(u+v) = (12/13 * cos u) - ((-8/17) * sin v)
To find the value of cos u, we can use the Pythagorean identity:
cos^2 u + sin^2 u = 1
Solving for cos u:
cos^2 u = 1 - sin^2 u
cos u = sqrt(1 - sin^2 u)
cos u = sqrt(1 - (-8/17)^2)
cos u = sqrt(1 - 64/289)
cos u = sqrt((289 - 64)/289)
cos u = sqrt(225/289)
cos u = 15/17
Substituting the values of cos u and sin v into cos(u+v):
cos(u+v) = (12/13 * 15/17) - ((-8/17) * sin v)
cos(u+v) = 180/221 - ((-8/17) * sin v)
Now, we need to determine the value of sin v. To do this, we can use the Pythagorean identity:
sin^2 v + cos^2 v = 1
Solving for sin v:
sin^2 v = 1 - cos^2 v
sin v = sqrt(1 - cos^2 v)
sin v = sqrt(1 - (12/13)^2)
sin v = sqrt(1 - 144/169)
sin v = sqrt((169 - 144)/169)
sin v = sqrt(25/169)
sin v = 5/13
Substituting the value of sin v into cos(u+v):
cos(u+v) = 180/221 - ((-8/17) * sin v)
cos(u+v) = 180/221 - ((-8/17) * 5/13)
cos(u+v) = 180/221 - (40/221)
cos(u+v) = 140/221
Therefore, the exact value of cos(u+v) is 140/221.
To find the exact value of cos(u+v), we'll need to use trigonometric identities and the given information.
First, let's find sin v. We know that sin^2(v) + cos^2(v) = 1. Substituting cos v = 12/13, we have sin^2(v) + (12/13)^2 = 1. Solving for sin v, we get sin v = ± (5/13).
Since cos(u+v) = cos u * cos v - sin u * sin v, we can substitute the given values to find cos(u+v).
cos(u+v) = (cos u) * (cos v) - (sin u) * (sin v)
Substituting sin u = -8/17, cos v = 12/13, and sin v = ± (5/13), we have:
cos(u+v) = (cos u) * (12/13) - (-8/17) * (5/13)
To use cos u, we need to find cos u. We know that sin^2(u) + cos^2(u) = 1. Substituting sin u = -8/17, we have (-8/17)^2 + cos^2(u) = 1. Solving for cos u, we get cos u = ± (15/17).
cos(u+v) = (± 15/17) * (12/13) - (-8/17) * (5/13)
Now we can calculate the value:
cos(u+v) = (15/17)*(12/13) + (8/17)*(5/13)
Simplifying the expression:
cos(u+v) = 180/221 + 40/221
cos(u+v) = 220/221
Therefore, the exact value of cos(u+v) is 220/221.