Question

Mau tanya jika 3 sin x + 4 cos y = 5 maka nilai minimum 3 cos x + 4 sin y adalah .....

Answer 1

Coba dikuadratkan:
5² = (3sin(x) + 4cos(y))²
25 = 9sin²(x) + 24sin(x)cos(y) + 16cos²(y)
Identitas:
25 = 9(1-cos²(x)) + 24sin(x)cos(y) + 16(1-sin²(y))
25 = 9-9cos²(x) + 24sin(x)cos(y) + 16-16sin²(y)
25 = 25 + 24sin(x)cos(y) - (9cos²(x)+16sin²(y))
0 = 24sin(x)cos(y) - (9cos²(x)+16sin²(y))
Sehingga:
9cos²(x) - 24sin(x)cos(y) + 16sin²(y) = 0
Alias:
(3cos(x) - 4sin(y))² = 0
Didapat demikian:
3cos(x) - 4sin(y) = 0
Dan:
4sin(y) = 3cos(x)

Dan:
3cos(x) + 4sin(y)
= 3cos(x) + 3cos(x)
= 6cos(x)
Mengingat nilai minimum cos(x) = -1
Maka, nilai minimumnya adalah:
= 6(-1)
= -6