SOLUTION: Find the exact value of the trigonometric function below given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III). cos (u-v)

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Question 1163852: Find the exact value of the trigonometric function below given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III).
cos (u-v)
Found 3 solutions by solver91311, Tatiana_Stebko, MathTherapy:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Use the Cosine of the Difference of Two Angles Identity:



To do this you will have to calculate and given and . The process in general is:





(Pythagorean Identity)



(Choose sign based on quadrant)

Finding cosine given sine is the same process.

John

My calculator said it, I believe it, that settles it

Answer by Tatiana_Stebko(1539) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28u-v%29=cos+u%2Acosv%2Bsinu%2Asinv
%28sin++u%29%5E2%2B%28cos+u%29%5E2=1
%28-7%2F25%29%5E2%2B%28cos+u%29%5E2=1
49%2F625%2B%28cos+u%29%5E2=1
%28cos+u%29%5E2=1-49%2F625
%28cos+u%29%5E2=576%2F625
u is in Quadrant III, therefore cos+u=-sqrt%28576%2F625%29
cos+u=-24%2F25
%28sin++v%29%5E2%2B%28cos+v%29%5E2=1
%28sin++v%29%5E2%2B%28-4%2F5%29%5E2=1
%28sin++v%29%5E2%2B16%2F25=1
%28sin+v%29%5E2=1-16%2F25
%28sin+v%29%5E2=9%2F25
v is in Quadrant III, therefore sin+v=-sqrt%289%2F25%29
sin+v=-3%2F5

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!

Find the exact value of the trigonometric function below given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III).
cos (u-v)
Difference of 2 angles formula: cos (A  -  B) = cos A cos B + sin A sin B
This gives us: cos (u  -  v) = cos u cos v + sin u sin v

The above represents a "7-24-25" PYTHAG TRIPLE, and so, x = 24.
However, because the opposite side (y), and the adjacent side (x), are in the 3rd quadrant, then y = O = - 7, and x = A = - 24.
Therefore, 


The above represents a "3-4-5" PYTHAG TRIPLE, and so, x = 3.
However, because the opposite side (y), and the adjacent side (x), are in the 3rd quadrant, then y = O = - 4, and x = A = - 3.
Therefore, 

We now get: