Diketahui tan(theta)=2/3, di mana theta sudut lancip.

a. cot(2 theta) = 5/12, b. tan(315-theta) = -5

Pembahasan

Diketahui tan(theta) = 2/3, di mana theta adalah sudut lancip.

a. Menghitung cot(2 theta):
Rumus cot(2 theta) = (cot^2(theta) - 1) / (2 cot(theta))
Karena tan(theta) = 2/3, maka cot(theta) = 1 / tan(theta) = 1 / (2/3) = 3/2.
Substitusikan nilai cot(theta) ke dalam rumus:
cot(2 theta) = ((3/2)^2 - 1) / (2 * (3/2))
cot(2 theta) = (9/4 - 1) / 3
cot(2 theta) = (9/4 - 4/4) / 3
cot(2 theta) = (5/4) / 3
cot(2 theta) = 5/12

b. Menghitung tan(315 - theta):
Rumus tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
Di sini, A = 315 derajat dan B = theta.
Kita tahu tan(315) = tan(360 - 45) = -tan(45) = -1.
Substitusikan nilai tan(315) dan tan(theta) ke dalam rumus:
tan(315 - theta) = (tan(315) - tan(theta)) / (1 + tan(315) tan(theta))
tan(315 - theta) = (-1 - 2/3) / (1 + (-1) * (2/3))
tan(315 - theta) = (-3/3 - 2/3) / (1 - 2/3)
tan(315 - theta) = (-5/3) / (3/3 - 2/3)
tan(315 - theta) = (-5/3) / (1/3)
tan(315 - theta) = -5

Hasilnya adalah:
a. cot(2 theta) = 5/12
b. tan(315 - theta) = -5