Find the exact value of the trig expression given that sin u = -5/13 and cos v = -4/5.
u and v are in the second quadrant.
cos(v - u)
well, just remember that
sin = y/r
cos = x/r
sin u cannot be negative if u is in QII.
Anyway, just recall your common triangles: 5-12-13 and 3-4-5. Draw them in the proper quadrant and find the missing side.
Then just plug the numbers into your difference formula for cosine.
Thanks Steve
To find the value of cos(v - u), we need to know the values of cos u and sin v. We are given the values of sin u and cos v, so let's find sin v.
Since cos v = -4/5, we can use the Pythagorean identity to find sin v. The Pythagorean identity states that sin^2(v) + cos^2(v) = 1.
Plugging in the given value of cos v, we can solve for sin v:
sin^2(v) + (-4/5)^2 = 1
sin^2(v) + 16/25 = 1
sin^2(v) = 1 - 16/25
sin^2(v) = 25/25 - 16/25
sin^2(v) = 9/25
To find sin v, we take the square root of both sides:
sin v = ± √(9/25)
sin v = ± (3/5)
Since v is in the second quadrant, sin v is positive. Therefore, sin v = 3/5.
Now that we have the values of sin u, cos v, and sin v, we can find cos(v - u) using the difference of angles formula for cosine:
cos(v - u) = cos v * cos u + sin v * sin u
Substituting the given values:
cos(v - u) = (-4/5) * (-5/13) + (3/5) * (-5/13)
Simplifying:
cos(v - u) = 20/65 - 15/65
cos(v - u) = 5/65
Therefore, the exact value of cos(v - u) is 5/65.