Question

Jika tan α = -1/2 dan π/2 < α < π, hitunglah nilai dari :
a. Sin 4α
b. Cos 4α
c. Tan 4α

Answer 1

Untuk kuadran 2 :

sin x = positif
cos x = negatif
tan x = negatif

tan a = depan / samping

depan = 1
samping = 2

miring = √1² + 2²
= √1 + 4
= √5

sin a = depan / miring
= 1 / √5
cos a = samping / miring
= -2 / √5

==========

sin 4a = sin(2a + 2a)
= sin2a cos2a + sin2a cos2a
= 2(sin2a cos2a)
= 2(sin(a + a) cos(a + a))
= 2((sina cosa + sina cosa)(cosa cosa - sina sina))
= 2(2(sina cosa)(cos²a - sin²a))
= 2(2(1/√5)(-2/√5)((-2/√5)² - (1/√5)²))
= 2(2(-2/5)(4/5 - 1/5))
= 2(-4/5)(3/5)
= 2(-12/25)
= -24/25

cos4a = cos(2a + 2a)
= cos2a cos2a - sin2a sin2a
= cos(a + a) cos(a + a) - sin(a + a) sin(a + a)
= (cosa cosa - sina sina)(cosa cosa - sina sina) - (sina cosa + sina cosa)(sina cosa + sina cosa)
= (cosa cosa - sina sina)² - (sina cosa + sina cosa)²
= (cos²a - sin²a)² - (2(sina cosa))²
= ((-2/√5)² - (1/√5)²)² - (2(1/√5)(-2/√5))²
= (4/5 - 1/5)² - (2(-2/5))²
= (3/5)² - (-4/5)²
= 9/25 - 16/25
= -7/25

tan4a = sin4a / cos4a
= (-24/25) / (-7/25)
= 24/7