Question

Given sin(u) = 4/5 and cos(v) = -7/25. Find cos(u - v).?

Answer 1

`cos (u - v) = +-68/125, +-124/125`

Explanation:

`sin u = 4/5, cos u = -7/25`

`"To find " cos (u - v)`

`sin u = " opp. side / hyp." = 4/5`

`:. "Adj. side " = sqrt(5^2 - 4^2) = +-3`

`cos u = "Adj. side / Hyp." = +-3/5`

When the angle u is in I quadrant, cos u = + 3/5 and when in second quadrant, cos u = -3/5#

`"Similarly", sin v = sqrt(25^2 - 7^2) / 25 = +-24/25`

When the angle is in second quadrant, cosine is negative and sine is positive. Therefore, `sin v = 24/25`

When the angle is in third quadrant, cosine is negative as also sine. Therefore, `sin v = -24/25`

`cos (x - y) = cos u cos v + sin u sin v, " Identity"`

Case 1 ; u in I quadrant and v in II Quadrant :

`:. cos (u - v) = 4/5 * (-7/25) + (4/5 )* (24/25) = (-28 + 96) / 125 = 68/125`

Case 2 ; u in I quadrant and v in III Quadrant :

`:. cos (u - v) = 4/5 * (-7/25) + (4/5 )* (-24/25) = (-28 - 96) / 125 = -124/125`

Case 3 ; u in II quadrant and v in II Quadrant :

`:. cos (u - v) = -4/5 * (-7/25) + (4/5 )* (24/25) = (28 + 96) / 125 = 124/125`

Case 2 ; u in II quadrant and v in III Quadrant :

`:. cos (u - v) =- 4/5 * (-7/25) + (4/5 )* (-24/25) = (+28 - 96) / 125 = -68/125`