Diketahui sin x=54, dengan 0∘<x<90∘. Nilai cos 3x+cos x=...

Pertanyaan

Diketahui $\sin\;x=\frac45$ , dengan $0^\circ<x<90^\circ$ . Nilai $\cos\;3x+\cos\;x=...$

M. Nasrullah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

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Pembahasan

Ingat kembali:

$cos space A plus cos space B equals 2 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses$

cos space 2 x equals 1 minus 2 sin squared x

Aturan Pythagoras:

Perbandingan sisi pada trigonometri:

$sin space x equals fraction numerator sisi space depan space sudut over denominator sisi space miring end fraction cos space x equals fraction numerator sisi space samping space sudut over denominator sisi space miring end fraction$

Pertama kita tentukan terlebih dahulu cos space x :

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight x end cell equals cell 4 over 5 end cell row cell fraction numerator sisi space depan space sudut over denominator sisi space miring end fraction end cell equals cell 4 over 5 end cell row blank rightwards arrow cell sisi space depan space sudut equals 4 end cell row blank rightwards arrow cell sisi space miring equals 5 end cell end table$

Dengan menggunakan teorema Pythagoras:

table attributes columnalign right center left columnspacing 0px end attributes row cell sisi space samping end cell equals cell square root of kuadrat space sisi space miring minus kuadrat space sisi space depan end root end cell row blank equals cell square root of 5 squared minus 4 squared end root end cell row blank equals cell square root of 9 end cell row blank equals 3 end table

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight x end cell equals cell fraction numerator sisi space samping space sudut over denominator sisi space miring end fraction end cell row blank equals cell 3 over 5 end cell end table$

Sehingga diperoleh perhitungan:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x plus cos space x end cell equals cell 2 space cos space open parentheses fraction numerator 3 x plus x over denominator 2 end fraction close parentheses times cos space open parentheses fraction numerator 3 x minus 1 over denominator 2 end fraction close parentheses end cell row blank equals cell 2 space cos space 2 x times cos space x end cell row blank equals cell open parentheses 1 minus 2 sin squared x close parentheses times cos space x end cell row blank equals cell open parentheses 1 minus 2 open parentheses 4 over 5 close parentheses squared close parentheses times 3 over 5 end cell row blank equals cell open parentheses 1 minus 32 over 25 close parentheses times 3 over 5 end cell row blank equals cell open parentheses fraction numerator 25 minus 32 over denominator 25 end fraction close parentheses times 3 over 5 end cell row blank equals cell fraction numerator negative 7 over denominator 25 end fraction times 3 over 5 end cell row blank equals cell negative 21 over 125 end cell end table$

Dengan mikian, nilai cos space 3 x plus cos space x adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 21 over 125 end cell end table

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