Question

Find the exact value of the trigonometric expression given that sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17, where 0 < v < π/2. ? cos(u−v)

Find the exact value of the trigonometric expression given that
sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17,
where 0 < v < π/2. ?

cos(u−v)

Answer 1

`cos (u - v) = 36/85`

Explanation:

Trig identity:
cos (u - v) = cos u.cos v + sin u.sin v (1)
`sin u = - 3/5`. Find cos u
`cos^2 u = 1 - sin^2 u = 1 - 9/25 = 16/25`
`cos u = 4/5` (because u lies in Quadrant 4)
`cos v = 15/17`. Find sin v.
`sin^2 v = 1 - cos^2 v = 1 - 225/289 = 64/289`
`sin v = 8/17` (because v lies in Quadrant 1)
Replace these above values into identity (1)
`cos (u - v) = (15/17)(4/5) + (-3/5)(8/17) = 36/85`