Pertanyaan

Jika nilai sin A = 13 12 maka nilai sin 2 A , cos 2 A dan tan 2 A adalah ...

Jika nilai $sin A = \frac{12}{13}$ maka nilai $sin 2 A, cos 2 A$ dan $tan 2 A$ adalah ...

L. Rante

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

Jawaban terverifikasi

Jawaban

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sin space 2 straight A equals 120 over 169 comma space cos space 2 straight A equals negative 119 over 169 space dan space tan space 2 straight A equals negative 120 over 119

Pembahasan

Berdasarkan defenisi sinus, . Karena maka Dengan menggunakan teorema Pythagoras, diperoleh Sehingga Dengan menggunakan rumus sudut rangkap, diperoleh Jadi, .

Berdasarkan defenisi sinus, $sin space straight A equals fraction numerator panjang space sisi space depan space angle straight A over denominator panjang space sisi space miring end fraction$ . Karena sin space straight A equals 12 over 13 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell panjang space sisi space depan space angle straight A end cell equals 12 row cell panjang space sisi space miring end cell equals 13 end table

Dengan menggunakan teorema Pythagoras, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell panjang space sisi space samping space angle straight A space end cell equals cell square root of 13 squared minus 12 squared end root end cell row blank equals cell square root of 169 minus 144 end root end cell row blank equals cell square root of 25 end cell row blank equals 5 end table

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight A end cell equals cell fraction numerator panjang space sisi space samping space angle straight A over denominator panjang space sisi space miring end fraction equals 5 over 13 space end cell row cell tan space straight A end cell equals cell fraction numerator panjang space sisi space depan space angle straight A over denominator panjang space sisi space samping space angle straight A end fraction equals 12 over 5 end cell end table$

Dengan menggunakan rumus sudut rangkap, diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 A end cell equals cell 2 times sin space A times cos space A end cell row blank equals cell 2 times 12 over 13 times 5 over 13 end cell row blank equals cell 120 over 169 end cell row blank blank blank row cell cos space 2 A end cell equals cell cos squared A minus sin squared A end cell row blank equals cell open parentheses 5 over 13 close parentheses squared minus open parentheses 12 over 13 close parentheses squared end cell row blank equals cell 25 over 169 minus 144 over 169 end cell row blank equals cell negative 119 over 169 end cell row blank blank blank row cell tan space 2 A end cell equals cell fraction numerator 2 times tan space A over denominator 1 minus tan squared A end fraction end cell row blank equals cell fraction numerator 2 times open parentheses begin display style 12 over 5 end style close parentheses over denominator 1 minus open parentheses begin display style 12 over 5 end style close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style 24 over 5 end style over denominator 1 minus begin display style 144 over 25 end style end fraction end cell row blank equals cell fraction numerator begin display style 24 over 5 end style over denominator negative begin display style 119 over 25 end style end fraction end cell row blank equals cell negative fraction numerator 24 cross times 25 over denominator 5 cross times 119 end fraction end cell row blank equals cell negative 120 over 119 end cell end table$

Jadi, sin space 2 straight A equals 120 over 169 comma space cos space 2 straight A equals negative 119 over 169 space dan space tan space 2 straight A equals negative 120 over 119 .