Diketahui alpha di kuadran III dan beta di kuadran II. Jika

sin(a+b)=16/65, sin(a-b)=56/65, cos(a+b)=63/65, cos(a-b)=33/65, tan(a+b)=16/63, tan(a-b)=56/33

Pembahasan

Diketahui:
Alpha di kuadran III, cos a = -4/5
Beta di kuadran II, tan b = -5/12

Langkah 1: Cari nilai sin a, tan a, sin b, cos b.

Untuk Alpha (kuadran III):
cos a = -4/5 (samping/miring). Karena di kuadran III, kedua sisi (x dan y) negatif.
Sisi depan (y) = -sqrt(miring^2 - samping^2) = -sqrt(5^2 - (-4)^2) = -sqrt(25 - 16) = -sqrt(9) = -3.
Jadi, sin a = -3/5, tan a = sin a / cos a = (-3/5) / (-4/5) = 3/4.

Untuk Beta (kuadran II):
tan b = -5/12 (depan/samping). Karena di kuadran II, sisi depan (y) positif dan sisi samping (x) negatif.
Sisi samping (x) = -12.
Sisi depan (y) = 5.
Miring = sqrt(samping^2 + depan^2) = sqrt((-12)^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13.
Jadi, sin b = 5/13, cos b = -12/13.

Langkah 2: Hitung nilai-nilai yang diminta menggunakan rumus penjumlahan dan pengurangan.

1.  **sin(a + b)** = sin a cos b + cos a sin b
    = (-3/5)(-12/13) + (-4/5)(5/13)
    = 36/65 - 20/65
    = 16/65

2.  **sin(a - b)** = sin a cos b - cos a sin b
    = (-3/5)(-12/13) - (-4/5)(5/13)
    = 36/65 + 20/65
    = 56/65

3.  **cos(a + b)** = cos a cos b - sin a sin b
    = (-4/5)(-12/13) - (-3/5)(5/13)
    = 48/65 + 15/65
    = 63/65

4.  **cos(a - b)** = cos a cos b + sin a sin b
    = (-4/5)(-12/13) + (-3/5)(5/13)
    = 48/65 - 15/65
    = 33/65

5.  **tan(a + b)** = (tan a + tan b) / (1 - tan a tan b)
    = (3/4 + (-5/12)) / (1 - (3/4)(-5/12))
    = (9/12 - 5/12) / (1 + 15/48)
    = (4/12) / (48/48 + 15/48)
    = (1/3) / (63/48)
    = (1/3) * (48/63)
    = 16/63

6.  **tan(a - b)** = (tan a - tan b) / (1 + tan a tan b)
    = (3/4 - (-5/12)) / (1 + (3/4)(-5/12))
    = (9/12 + 5/12) / (1 - 15/48)
    = (14/12) / (48/48 - 15/48)
    = (7/6) / (33/48)
    = (7/6) * (48/33)
    = 7 * 8 / 33
    = 56/33

Jadi, sin(a+b)=16/65, sin(a-b)=56/65, cos(a+b)=63/65, cos(a-b)=33/65, tan(a+b)=16/63, tan(a-b)=56/33.