Find the exact value of cos(u + v) ) given that sin u= 9/41 cos nu=- 15/17 . (Both u and v are in Quadrant II.)
in QII
sinu = 9/41
cosu = -40/41
cosv = -15/17
sinv = 8/17
cos(u+v) = cosu cosv - sinu sinv
so plug in your numbers
Answer: cos(u+v)=528/697
To find the exact value of cos(u + v), we need to use the trigonometric identities. In this case, we're given sin(u) = 9/41 and cos(v) = -15/17, and both u and v are in Quadrant II.
First, let's find cos(u) using the Pythagorean identity:
cos^2(u) = 1 - sin^2(u)
cos^2(u) = 1 - (9/41)^2
cos^2(u) = 1 - 81/1681
cos^2(u) = (1681 - 81)/1681
cos^2(u) = 1600/1681
cos(u) = ±√(1600/1681)
Since u is in Quadrant II, cos(u) is negative. Therefore:
cos(u) = -√(1600/1681)
cos(u) = -40/41
Now, let's find sin(v) using the Pythagorean identity:
sin^2(v) = 1 - cos^2(v)
sin^2(v) = 1 - (-15/17)^2
sin^2(v) = 1 - 225/289
sin^2(v) = (289 - 225)/289
sin^2(v) = 64/289
sin(v) = ±√(64/289)
Since v is in Quadrant II, sin(v) is positive. Therefore:
sin(v) = √(64/289)
sin(v) = 8/17
Now, we can use the addition formula for cosine:
cos(u + v) = cos(u)cos(v) - sin(u)sin(v)
cos(u + v) = (-40/41)(-15/17) - (9/41)(8/17)
cos(u + v) = (600/697) - (72/697)
cos(u + v) = 528/697
Therefore, the exact value of cos(u + v) is 528/697.
To find the exact value of cos(u + v), we can use the cosine addition formula:
cos(u + v) = cos(u)cos(v) - sin(u)sin(v)
To find cos(u) and sin(u), we can use the given information that u is in Quadrant II and sin(u) = 9/41 and cos(u) = -15/17.
Since u is in Quadrant II, sin(u) is positive and cos(u) is negative. Therefore, sin(u) = 9/41 and cos(u) = -15/17.
To find cos(v), we need additional information.