Nilai dari sec 40 + sec 80 sec 160=...

Nilai dari sec 40° + sec 80° + sec 160° adalah 6.

Pembahasan

Untuk menghitung nilai sec(40°) + sec(80°) + sec(160°):
Kita gunakan identitas sec(θ) = 1/cos(θ).
Nilai = 1/cos(40°) + 1/cos(80°) + 1/cos(160°)
Samakan penyebutnya:
= [cos(80°)cos(160°) + cos(40°)cos(160°) + cos(40°)cos(80°)] / [cos(40°)cos(80°)cos(160°)]
Gunakan identitas 2cosAcosB = cos(A+B) + cos(A-B).
Pembilang:
cos(80°)cos(160°) = 1/2 [cos(240°) + cos(-80°)] = 1/2 [cos(240°) + cos(80°)]
cos(40°)cos(160°) = 1/2 [cos(200°) + cos(-120°)] = 1/2 [cos(200°) + cos(120°)]
cos(40°)cos(80°) = 1/2 [cos(120°) + cos(-40°)] = 1/2 [cos(120°) + cos(40°)]

Jumlahkan:
1/2 [cos(240°) + cos(80°) + cos(200°) + cos(120°) + cos(120°) + cos(40°)]
= 1/2 [cos(240°) + cos(80°) + cos(200°) + 2cos(120°) + cos(40°)]
Kita tahu cos(120°) = -1/2 dan cos(240°) = -1/2.
= 1/2 [-1/2 + cos(80°) + cos(200°) - 1 + cos(40°)]
= 1/2 [-3/2 + cos(80°) + cos(200°) + cos(40°)]

Gunakan cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2).
cos(80°) + cos(40°) = 2cos(60°)cos(20°) = 2(1/2)cos(20°) = cos(20°)
cos(200°) = cos(180° + 20°) = -cos(20°)
Jadi, cos(80°) + cos(200°) + cos(40°) = cos(20°) - cos(20°) = 0.

Pembilang = 1/2 [-3/2 + 0] = -3/4.

Penyebut:
cos(40°)cos(80°)cos(160°)
= 1/2 [cos(120°) + cos(40°)] cos(160°)
= 1/2 [-1/2 + cos(40°)] cos(160°)
= -1/4 cos(160°) + 1/2 cos(40°)cos(160°)
= -1/4 cos(160°) + 1/2 * 1/2 [cos(200°) + cos(120°)]
= -1/4 cos(160°) + 1/4 [cos(200°) - 1/2]
= -1/4 cos(160°) + 1/4 cos(200°) - 1/8
cos(160°) = cos(180°-20°) = -cos(20°)
cos(200°) = cos(180°+20°) = -cos(20°)
= -1/4 (-cos(20°)) + 1/4 (-cos(20°)) - 1/8
= 1/4 cos(20°) - 1/4 cos(20°) - 1/8
= -1/8

Nilai = Pembilang / Penyebut = (-3/4) / (-1/8) = (-3/4) * (-8/1) = 6.

*Perhitungan alternatif menggunakan identitas lain*
Nilai dari sec 40 + sec 80 + sec 160 = 1/cos 40 + 1/cos 80 + 1/cos 160
= (cos 80 cos 160 + cos 40 cos 160 + cos 40 cos 80) / (cos 40 cos 80 cos 160)
Pembilang: cos 80 cos 160 + cos 40 cos 160 + cos 40 cos 80
= cos 80 cos(180-20) + cos 40 cos(180-20) + cos 40 cos 80
= -cos 80 cos 20 - cos 40 cos 20 + cos 40 cos 80
= -cos 20 (cos 80 + cos 40) + cos 40 cos 80
= -cos 20 (2 cos 60 cos 20) + cos 40 cos 80
= -cos 20 (2 * 1/2 * cos 20) + cos 40 cos 80
= -cos^2 20 + cos 40 cos 80
= -(1-sin^2 20) + 1/2 (cos 120 + cos 40)
= -1 + sin^2 20 + 1/2 (-1/2 + cos 40)
= -1 + sin^2 20 - 1/4 + 1/2 cos 40
= -5/4 + sin^2 20 + 1/2 cos 40
Karena cos 40 = 1 - 2 sin^2 20, maka sin^2 20 = (1 - cos 40)/2
= -5/4 + (1 - cos 40)/2 + 1/2 cos 40
= -5/4 + 1/2 - 1/2 cos 40 + 1/2 cos 40
= -5/4 + 2/4 = -3/4

Penyebut: cos 40 cos 80 cos 160
= cos 40 cos 80 cos (180-20)
= -cos 40 cos 80 cos 20
= -cos 20 * cos 40 * cos 80
= -cos 20 * 1/2 (cos 120 + cos 40)
= -cos 20 * 1/2 (-1/2 + cos 40)
= -1/2 cos 20 (-1/2 + cos 40)
= 1/4 cos 20 - 1/2 cos 20 cos 40
= 1/4 cos 20 - 1/2 * 1/2 (cos 60 + cos 20)
= 1/4 cos 20 - 1/4 (1/2 + cos 20)
= 1/4 cos 20 - 1/8 - 1/4 cos 20
= -1/8

Nilai = (-3/4) / (-1/8) = 6.