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KELAS : KELOMPOK: ANGGOTA KELOMPOK: 1.. 2. 3. 4. 5.
SMA KELAS XI PEMINATAN Alokasi Waktu: 2 x 45 Menit
LKPD
LEMBAR KEGIATAN PESERTA DIDIK
PERSAMAAN TRIGONOMETRI (BENTUK KUADRAT)
KOMPETENSI DASAR 3.1 Menjelaskan dan menentukan penyelesaian persamaan trigonometri 4.1 Memodelkan dan Menyelesaikan masalah yang berkaitan dengan persamaan trigonometri
INDIKATOR PENCAPAIAN KOMPETENSI
3.1.4. Menentukan penyelesaian persamaan trigonometri bentuk kuadrat 4.1.2. Menyelesaikan masalah yang berkaitan dengan persamaan trigonometri bentuk kuadrat
PERSAMAAN TRIGONOMETRI (BENTUK KUADRAT) Mengamati
Waktu : 5 Menit
Ingat Kembali
Amati persamaan trigonometri pada tabel berikut ini. 1 π π ππ2 π₯ + π sin π₯ + π = 0 Misalkan : sin π₯ = π Maka, π π ππ2 π₯ + π sin π₯ + π = 0 π π2 + π π + π = 0 ........................(persamaan 1) 2 π πππ 2 π₯ + π cos π₯ + π = 0 Misalkan : cos π₯ = π Maka, π πππ 2 π₯ + π cos π₯ + π = 0 π π 2 + π π + π = 0 ........................(persamaan 2) 3 π π‘ππ2 π₯ + π tan π₯ + π = 0 Misalkan : tan π₯ = π Maka, π π‘ππ2 π₯ + π tan π₯ + π = 0 π π 2 + π π + π = 0 ........................(persamaan 3)
Rumus- Rumus Trigonometri 1
csc π₯ = sin π₯ 1
sec π₯ = cos π₯ tan π₯ =
sin π₯ cos π₯
π ππ2 π₯ + πππ 2 π₯ = 1 1 + π‘ππ2 π₯ = π ππ 2 π₯ πππ‘ 2 π₯ + 1 = ππ π 2 π₯ sin 2π₯ = 2 sin π₯ cos π₯ cos 2π₯ = 1 β 2 π ππ2 π₯ cos 2π₯ = 2πππ 2 π₯ β 1 cos 2π₯ = πππ 2 π₯ β π ππ2 π₯ 2 tan π₯
tan 2π₯ = 1βπ‘ππ2 π₯ sin 3π₯ = 3 sin π₯ β 4π ππ3 π₯
Perhatikan bahwa persamaan 1, 2, dan 3, adalah persamaan kuadrat. Dapatkah kalian menentukan nilai sudut yang memenuhi persamaan-
cos 3π₯ = 4πππ 3 π₯ β 3 cos π₯ (sin π₯ β cos π₯)2 = 1 β 2 sin π₯ cos π₯
persamaan tersebut? Bagaimanakah cara menentukan sudut yang memenuhi persamaan tersebut?
Menanya
Waktu : 5 Menit
Coba kalian buat pertanyaan-pertanyaan yang bisa kalian temukan pada kegiatan mengamati tersebut ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................ ................................................................................
Mengumpulkan Informasi
Waktu : 5 Menit
Carilah berbagai informasi yang kalian butuhkan untuk menemukan jawaban dari pertanyaan yang kalian buat dari berbagai sumber. ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ......................................................................................... ..........................
Mengasosiasikan Kegiatan 1 : Menentukan penyelesaian persamaan trigonometri bentuk kuadrat
Waktu : 20 Menit
1. Tentukanlah nilai x yang memenuhi persamaan 2 πππ 2 π₯ β 5 cos π₯ + 2 = 0 untuk 0Β° β€ π₯ β€ 360Β° Penyelesaian : Misalkan : cos π₯ = β―, maka : 2 πππ 2 π₯ β 5 cos π₯ + 2 = 0 2 (β¦ )2 β 5(β¦ ) + 2 = 0 (β¦ β¦ β¦ )(β¦ β¦ β¦ ) = 0
Karena : β¦ = cos π₯, maka : cos π₯ = β― atau
cos π₯ = β―
π₯ = β― + π. 360Β° Atau π₯ = β β― Β° + π. 360Β°
π₯3 = β―
Untuk k = ..., maka : (β¦ β¦ β¦ ) = 0 atau β¦ = β― atau
(β¦ β¦ β¦ ) = 0 β¦=β―
π₯1 = β― Β° + (0). 360Β° = β― Β° π₯2 = β― Β° + (β¦ ). 360Β° β¦ Β°
Maka, himpunan penyelesaian yang memenuhi persamaan 2 πππ 2 π₯ β 5 cos π₯ + 2 = 0 untuk 0Β° β€ π₯ β€ 360Β° adalah {β¦ Β°, β¦ Β°}
2. Tentukanlah nilai x yang memenuhi persamaan sin π₯ β 2 π ππ 2 π₯ = 0 untuk 0Β° β€ π₯ β€ 720Β° Untuk k = 0 Penyelesaian : π₯1 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ Misalkan : sin π₯ = β―, maka : ................................................................................................................ π₯2 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ ................................................................................................................ π₯3 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ ................................................................................................................ π₯4 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ Untuk k = 1 ................................................................................................................ π₯5 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ ................................................................................................................ π₯ = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ 6 ................................................................................................................ π₯ = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ 7 ................................................................................................................ π₯8 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ ................................................................................................................ Untuk k = 2 ................................................................................................................ π₯9 = β― β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ Β° ................................................................................................................ ................................................................................................................ Maka, himpunan penyelesaian yang memenuhi persamaan 2 ................................................................................................................ sin π₯ β 2 π ππ π₯ = 0 untuk 0Β° β€ π₯ β€ 720Β°adalah ................................................................................................................ {β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . } ................................................................................................................ 3. Tentukanlah nilai x yang memenuhi persamaan π‘ππ 2 π β 2 tan π = β1 untuk 0 β€ π₯ β€ 6π
Penyelesaian :
Kegiatan 2 : Menentukan penyelesaian persamaan trigonometri bentuk kuadrat (Lanjutan)
Waktu : 20 Menit
4. Tentukanlah nilai x yang memenuhi persamaan πππ 2 πΌ β 2 π ππ 2 πΌ = 0 untuk 0Β° β€ π₯ β€ 360Β° Penyelesaian : πππ 2 πΌ β 2 π ππ 2 πΌ = 0 (β¦ β¦ β¦ β¦ ) β 2 π ππ 2 πΌ = 0 β¦β¦β¦β¦β¦β¦ = 0 Misalkan : β¦ β¦ = β― β¦β¦β¦β¦β¦β¦ = 0 β¦β¦β¦β¦ = 0 β¦ β¦2 = β¦ β¦
β¦ = Β±ββ¦ β¦ = Β±β―
Karena : β¦ = β― β¦ β¦β¦β¦ = β― atau β¦β¦β¦ = β― (πΊπ’πππππ π‘ππππ π’ππ‘π’π ππππππ‘π’πππ πππππ .... yang bernilai β¦ β¦ β¦ ) β¦β¦β¦ = β― πΌ = β― Β° β¦β² + π. 360Β°
atau
β¦β¦β¦ = β― πΌ = (β¦ Β° + β― Β° β¦β² ) + π. 360Β° = β― Β° β¦β² + π. 360Β° πΌ = (β¦ Β° β β― Β° β¦ β²) + π. 360Β° = β― Β° β¦ β² + π. 360Β°
Untuk k = 0
πΌ1 = 35Β°16β² + (0). 360Β° = 35Β°16β² πΌ2 = 215Β°16β² + (0). 360 = 215Β°16β² πΌ3 = 324Β°44β² + (0). 360Β° = 324Β°44β²
Maka, himpunan penyelesaian yang memenuhi persamaan πππ 2 πΌ β 2 π ππ 2 πΌ = 0 untuk 0Β° β€ π₯ β€ 360Β° adalah {35Β°16β² , 215Β°16β² , 324Β°44β²}
5. Tentukanlah nilai x yang memenuhi persamaan 3 + cos 2π₯ = 8 cos π₯ untuk 0Β° β€ π₯ β€ 180Β° Penyelesaian : 3 + cos 2π₯ = 8 cos π₯ β¦ + (β¦ β¦ β¦ β¦ β¦ β¦ ) = β― β¦β¦β¦β¦β¦β¦β¦β¦β¦ = 0 Misalkan : cos π₯ = β― , maka : 2 πππ 2 π₯ β 8 cos π₯ + 2 = 0 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ = 0 π1,2 =
βπΒ±βπ2 β4ππ 2π
π1,2 =
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.. β¦β¦β¦β¦β¦..
π1,2 =
β¦β¦β¦β¦β¦β¦β¦β¦β¦. β¦β¦β¦..
π1,2 = β― β¦ β¦ β¦ β¦ .. π1,2 = β― β¦ β¦ β¦ β¦ .. π1 = β― β¦ β¦ β¦ β¦ .. π2 = β― β¦ β¦ β¦ β¦ ..
Karena : π = cos π₯ atau
cos π₯ = β―
cos π₯ = β― π₯ = β―β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦
(πΊπ’πππππ π‘ππππ π’ππ‘π’π ππππππ‘π’πππ πππππ cos x yang bernilai .........) cos β¦ Β° β¦β² = β― (π1 ) cos β¦ Β° β¦β² = β― (π2 ) π1 = (β¦ β¦ β¦ ) β (β¦ β¦ β¦ ) = (β¦ β¦ β¦ ) (Selisih π1 dan π1 ) π2 = (β¦ β¦ β¦ ) β (β¦ β¦ β¦ ) = (β¦ β¦ β¦ ) (Selisih π2 dan π1 ) π=
60Γπ1 π2
=
60Γβ¦β¦β¦ β¦β¦β¦ β²
= β― β²β²
πππ π₯ = cos β¦ Β° β¦ β¦ β²β² π₯ = β― Β° β¦β² β¦ β²β² + π. 360Β° Untuk k = 0 π₯1 = β― Β° β¦β² β¦β²β² + (β¦ ). 360Β° = β― Β° β¦β² β¦ β²β²
Maka, himpunan penyelesaian yang memenuhi persamaan 3 + cos 2π₯ = 8 cos π₯ untuk 0Β° β€ π₯ β€ 180Β°adalah {β¦ Β° β¦β² β¦ β²β²}
KESIMPULAN Langkah dalam menyelesaikan persamaan trigonometri bentuk kuadrat yakni : ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ...................................................................................................................................................................................... ........................................................