{"id":281687,"date":"2020-04-17T14:57:14","date_gmt":"2020-04-17T07:57:14","guid":{"rendered":"https:\/\/quipperhome.wpcomstaging.com\/?p=281687"},"modified":"2022-03-01T17:58:55","modified_gmt":"2022-03-01T10:58:55","slug":"aturan-sinus-matematika-kelas-10","status":"publish","type":"post","link":"https:\/\/quipperhome.wpcomstaging.com\/mapel\/matematika\/aturan-sinus-matematika-kelas-10\/","title":{"rendered":"Aturan Sinus – Matematika Kelas 10"},"content":{"rendered":"
\"Aturan

Foto: pixabay.com<\/p><\/div>\n

Hai Quipperian, <\/span>what\u2019s up<\/span><\/em>? Semoga selalu sehat dan tetap semangat belajar bareng Quipper Blog, ya!<\/span><\/p>\n

Eropa merupakan salah satu benua yang memiliki banyak situs bersejarah. Salah satu situs bersejarah yang bisa dijadikan destinasi wisata adalah Menara Pisa di Italia. Menara Pisa merupakan salah satu menara yang cukup populer di dunia karena keunikannya, yaitu konstruksi gedungnya yang miring.\u00a0<\/span><\/p>\n

Sebenarnya, desain awal gedung tidaklah miring. Oleh karena struktur tanahnya rentan, maka menara otomatis menjadi miring. Membahas kemiringan, tentu tidak lepas dari besaran yang disebut sudut.\u00a0<\/span><\/p>\n

Kabar baiknya, kamu bisa mengukur sudut kemiringan Menara Pisa, <\/span>lho<\/span><\/em>. Bagaimana caranya? Yaitu menggunakan aturan sinus dan cosinus. Apa itu? Daripada penasaran, <\/span>check this out<\/span><\/em>!<\/span><\/p>\n

Aturan Sinus<\/strong><\/h2>\n
\"\"

Foto: pixabay.com<\/p><\/div>\n

Aturan sinus adalah perbandingan antara setiap sisi dan sinus sudut di depan sisi tersebut memiliki nilai yang sama. Aturan sinus ini berlaku pada segitiga, baik segitiga siku-siku maupun segitiga sembarang. <\/span><\/p>\n

Misalnya pada segitiga ABC yang memiliki panjang sisi <\/span>a<\/span><\/em>, <\/span>b<\/span><\/em>, dan <\/span>c<\/span><\/em>, serta sudut A, B, C, maka aturan sinus yang berlaku adalah sebagai berikut.<\/span><\/p>\n

\"\"<\/p>\n

Substitusikan nilai <\/span>h<\/span><\/em> pada persamaan (1) ke ke persamaan (2).<\/span><\/p>\n

\"\"<\/p>\n

Langkah di atas juga berlaku saat kamu akan mencari nilai <\/span>c<\/span><\/em>. Dengan demikian, aturan sinusnya menjadi seperti berikut.<\/span><\/p>\n

\"\"<\/p>\n

Aturan sinus bisa digunakan pada dua pasang sudut sisi yang saling berhadapan, di mana salah satunya tidak diketahui.<\/span><\/p>\n

Untuk lebih jelasnya, simak contoh soal berikut.<\/span><\/p>\n

Contoh Soal 1<\/strong><\/h3>\n

Pada awalnya, Menara Pisa dibangun dengan ketinggian 56 m. Ternyata, tanah di lokasi pembangunan menara rentan akan kerapuhan, sehingga terjadi kemiringan. Pada jarak 44 m dari dasar menara diperoleh sudut elevasi 55<\/span>o<\/span><\/sup>, tentukan derajat kemiringan menara dari posisi awalnya!<\/span><\/p>\n

Pembahasan:<\/strong><\/p>\n

Jika digambarkan akan menjadi seperti berikut.<\/span><\/p>\n

\"\"<\/p>\n

Persoalan di atas bisa disederhanakan menjadi segitiga berikut.<\/span><\/p>\n

\"\"<\/p>\n

Gunakan aturan sinus untuk menyelesaikan masalah di atas.<\/span><\/p>\n

\"\"<\/p>\n

Dengan demikian, besar sudut A = 180<\/span>o<\/span><\/sup> \u2013 (B + C)\u00a0 atau sudut A = 85<\/span>o<\/span><\/sup><\/p>\n

Jadi, derajat kemiringan Menara Pisa adalah <\/span>x<\/span><\/em> = 90<\/span>o<\/span><\/sup> \u2013 85<\/span>o<\/span><\/sup> = 5<\/span>o<\/span><\/sup>.<\/span><\/p>\n

Setelah kamu belajar tentang aturan sinus, kini saatnya belajar tentang aturan kosinus.<\/span><\/p>\n

Aturan Kosinus<\/strong><\/h2>\n
\"\"

Foto: pixabay.com<\/p><\/div>\n

Aturan kosinus adalah hubungan antara panjang sisi segitiga dan nilai kosinus dari salah satu sudut yang ada di dalam segitiga tersebut. Untuk mencari tahu persamaan yang berlaku di aturan kosinus, perhatikan gambar berikut.<\/span><\/p>\n

\"\"<\/p>\n

Pada segitiga ADC, berlaku:<\/span><\/p>\n

h<\/span><\/em>2<\/span><\/sup> = <\/span>b<\/span><\/em>2<\/span><\/sup> \u2013 AD<\/span>2<\/span><\/sup> \u2026(1)<\/span><\/p>\n

Pada segitiga siku-siku BDC, berlaku:<\/span><\/p>\n

h<\/span><\/em>2<\/span><\/sup> = <\/span>a<\/span><\/em>2<\/span><\/sup> \u2013 BD<\/span>2<\/span><\/sup> \u2026(2)<\/span><\/p>\n

Substitusikan nilai <\/span>h<\/span><\/em>2<\/span> pada persamaan (1) ke persamaan (2), diperoleh:<\/span><\/p>\n

b<\/span><\/em>2<\/span><\/sup> \u2013 AD<\/span>2<\/span><\/sup> = <\/span>a<\/span><\/em>2<\/span><\/sup> \u2013 BD<\/span>2<\/span><\/sup><\/p>\n

Oleh karena BD = <\/span>c<\/span><\/em> \u2013 AD, maka:<\/span><\/p>\n

b<\/span><\/em>2<\/span><\/sup> \u2013 AD<\/span>2<\/span><\/sup> = <\/span>a<\/span><\/em>2<\/span><\/sup> \u2013 (<\/span>c <\/span><\/em>\u2013 AD)<\/span>2<\/span><\/sup><\/p>\n

b<\/span><\/em>2<\/span><\/sup> \u2013 AD<\/span>2<\/span><\/sup> = <\/span>a<\/span><\/em>2<\/span><\/sup> \u2013 <\/span>c<\/span><\/em>2<\/span><\/sup> + 2.<\/span>c.<\/span><\/em>AD<\/span> \u2013 AD<\/span>2<\/span><\/sup><\/p>\n

a<\/span><\/em>2<\/span><\/sup> = <\/span>b<\/span><\/em>2<\/span><\/sup> + <\/span>c<\/span><\/em>2<\/span><\/sup> \u2013 2.<\/span>c<\/span><\/em>.AD<\/span><\/p>\n

 <\/p>\n

Pada segitiga ADC berlaku AD = <\/span>b<\/span><\/em>. cos A, sehingga dapat dinyatakan sebagai berikut.<\/span><\/p>\n

a<\/span><\/em>2<\/span><\/sup> = <\/span>b<\/span><\/em>2<\/span><\/sup> + <\/span>c<\/span><\/em>2<\/span><\/sup> \u2013 2.b.<\/span>c<\/span><\/em>.cos A<\/span><\/p>\n

Melalui cara yang sama, Quipperian akan mendapatkan nilai kosinus untuk panjang sisi lainnya (<\/span>b<\/span><\/em> dan <\/span>c<\/span><\/em>).<\/span><\/p>\n

Dari perhitungan di atas, aturan kosinus memenuhi persamaan berikut.<\/span><\/p>\n

\"\"<\/p>\n

Untuk lebih jelasnya tentang aturan kosinus, simak contoh soal berikut ini.<\/span><\/p>\n

Contoh Soal 2<\/strong><\/h3>\n

Kapal laut A dan B berlayar dari titik M pada waktu yang bersamaan. Kapal A berlayar dengan jurusan tiga angka 102<\/span>o<\/span><\/sup> dan kapal B berlayar dengan jurusan tiga angka 232<\/span>o<\/span><\/sup>. Jika kecepatan kapal A 30 km\/jam dan kecepatan kapal B 45 km\/jam, tentukan jarak kedua kapal tersebut setelah berlayar selama 3 jam!<\/span><\/p>\n

Pembahasan:<\/strong><\/p>\n

Untuk memudahkan Quipperian dalam menyelesaikan soal tersebut, perhatikan ilustrasi berikut.<\/span><\/p>\n

\"\"<\/p>\n

Oleh karena kecepatan A = 30 km\/jam, maka setelah 3 jam jarak yang ditempuh adalah 90 km. Untuk B, jarak yang ditempuh selama 3 jam adalah 135 km. Di sisi lain, besaran sudut AMB = 232<\/span>o<\/span><\/sup> -102<\/span>o<\/span><\/sup> = 130<\/span>o<\/span><\/sup>. Berdasarkan aturan kosinus, diperoleh:<\/span><\/p>\n

AB<\/span>2<\/span><\/sup> = AM<\/span>2<\/span><\/sup> + BM<\/span>2<\/span><\/sup> \u2013 2.AM.BM.cos M<\/span><\/p>\n

AB<\/span>2<\/span><\/sup> = 90<\/span>2<\/span><\/sup> + 135<\/span>2<\/span><\/sup> \u2013 2.90.135.cos 130<\/span>o<\/span><\/sup><\/p>\n

AB = 204,8 km<\/span><\/p>\n

Jadi, jarak kedua kapal tersebut setelah berlayar selama 3 jam adalah 204,8 km.<\/span><\/p>\n

Luas Segitiga<\/strong><\/h2>\n
\"\"

Foto: pexels.com<\/p><\/div>\n

Selama ini kamu telah belajar bagaimana cara mencari luas segitiga siku-siku, sama kaki, atau sama sisi. Lalu, bagaimana jika segitiganya bentuk sembarang atau panjang sisi-sisinya tidak sama? Ternyata, aturan sinus bisa kamu manfaatkan untuk mencari luas segitiga sembarang, <\/span>lho<\/span><\/em>. Berikut penjabarannya.<\/span><\/p>\n

Misalkan kamu memiliki segitiga sembarang ABC.<\/span><\/p>\n

\"\"\u00a0<\/span><\/p>\n

Pada segitiga ADC berlaku:<\/span><\/p>\n

\"\"<\/p>\n

Substitusikan nilai <\/span>h<\/span><\/em> pada persamaan (2) ke persamaan (1), diperoleh:<\/span><\/p>\n

\"\"<\/p>\n

Cara di atas bisa kamu gunakan untuk mencari luas segitiga sembarang.<\/span><\/p>\n

Untuk mengasah pemahamanmu tentang luas segitiga, simak contoh soal berikut.<\/span><\/p>\n

Contoh Soal 3<\/strong><\/h3>\n

Sebuah ruang tamu berukuran 15 kaki x 25 kaki dan memiliki bentuk seperti balok terbuka dengan atap yang membentuk segitiga pada sisi kanan dan kirinya seperti gambar berikut.<\/span><\/p>\n

\"\"<\/p>\n

Jika dua bagian dari atas membentuk sudut 50<\/span>o<\/span> dan 33<\/span>o<\/span> dengan bidang datar sebagaimana dalam gambar, tentukan luas dari segitiga yang terbentuk!<\/span><\/p>\n

Pembahasan:<\/strong><\/p>\n

Jika digambarkan kembali, segitiga yang terbentuk pada atap menjadi seperti berikut.<\/span><\/p>\n

\"\"<\/p>\n

Besar sudut C dapat dirumuskan sebagai berikut.<\/span><\/p>\n

C = 180<\/span>o<\/span><\/sup> \u2013 (33<\/span>o<\/span><\/sup> + 50<\/span>o<\/span><\/sup>)<\/span><\/p>\n

C = 97<\/span>o<\/span><\/sup><\/p>\n

Berdasarkan aturan sinus, diperoleh:<\/span><\/p>\n

\"\"<\/span><\/p>\n

Dengan demikian, luas segitiga yang terbentuk adalah:<\/span><\/p>\n

\"\"<\/p>\n

Jadi, luas segitiga yang terbentuk mendekati 49 kaki<\/span>2<\/span><\/sup>.<\/span><\/p>\n

Itulah pembahasan Quipper Blog kali ini tentang aturan sinus dan kosinus. Semoga Quipperian semakin paham, ya! Diharapkan materi ini bisa bermanfaat buat kamu semua.\u00a0<\/span><\/p>\n

Untuk mendapatkan materi lebih lengkap tentang aturan sinus dan kosinus, silahkan gabung bersama <\/span>Quipper Video<\/strong><\/a>. Jadikan Quipper Video sebagai mitra belajar yang menyenangkan. Salam Quipper!<\/span><\/p>\n

Penulis: Eka Viandari<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Hai Quipperian, what\u2019s up? Semoga selalu sehat dan tetap semangat belajar bareng Quipper Blog, ya! 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