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

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

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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.

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Topik: Identitas Trigonometri
Section: Penjumlahan Dan Pengurangan Sudut, Identitas Sudut Ganda Dan Setengah

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