Question

Jika x+y=, dengan x,y>0 dan memenuhi cos(2x) + sin(y) = 0, maka nilai cos(y-x) = ...

Answer 1

Untuk x + y = π
Maka:
y = π - x

0 = cos(2x) + sin(y)

Identitas : cos(2x) = 1 - 2sin²(x)

0 = 1-2sin²(x) + sin(π-x)
0 = 1-2sin²(x) + sin(x)
2sin²(x) - sin(x) - 1 = 0
Faktorkan:
(2sin(x) + 1)(sin(x) - 1) = 0

Maka:
cos(y-x) = cos([π - x] - x)
cos(y-x) = cos(π - 2x)
cos(y-x) = -cos(2x)
cos(y-x) = -(1-2sin²(x))
cos(y-x) = 2sin²(x) - 1

Untuk sin(x) = -1/2
cos(y-x) = 2(-1/2)² - 1
cos(y-x) = 2(1/4) - 1
cos(y-x) = 1/2 - 1
cos(y-x) = -1/2

Untuk sin(x) = 1
cos(y-x) = 2(1)² - 1
cos(y-x) = 2 - 1
cos(y-x) = 1