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Diketahui  dan . Jika besar sudut PQR adalah α, maka nilai dari sin α adalah ....

Pertanyaan

Diketahui begin mathsize 14px style stack Q P with rightwards arrow on top equals open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses end style dan begin mathsize 14px style stack Q R with rightwards arrow on top equals open parentheses table row 3 row 1 row 2 end table close parentheses end style. Jika besar sudut PQR adalah α, maka nilai dari sin α adalah ....

  1. begin mathsize 14px style fraction numerator 11 square root of 3 over denominator 13 end fraction end style

  2. begin mathsize 14px style fraction numerator 11 square root of 13 over denominator 13 end fraction end style

  3. begin mathsize 14px style fraction numerator 3 square root of 19 over denominator 19 end fraction end style

  4. begin mathsize 14px style fraction numerator 11 square root of 399 over denominator 266 end fraction end style

  5. begin mathsize 14px style fraction numerator 3 square root of 2.527 end root over denominator 266 end fraction end style

A. Rizky

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Perhatikan bahwa vektor begin mathsize 14px style stack Q P with rightwards arrow on top space d a n space stack Q R with rightwards arrow on top end style dapat disketsakan sebagai berikut

Karena begin mathsize 14px style stack Q P with rightwards arrow on top equals open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses end style dan begin mathsize 14px style stack Q R with rightwards arrow on top equals open parentheses table row 3 row 1 row 2 end table close parentheses end style, maka didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack Q P with rightwards arrow on top times stack Q R with rightwards arrow on top end cell equals cell open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses times open parentheses table row 3 row 1 row 2 end table close parentheses end cell row blank equals cell 2 times 3 plus open parentheses negative 3 close parentheses times 1 plus 5 times 2 end cell row blank equals cell 6 minus 3 plus 10 end cell row blank equals 13 end table end style

Kemudian, kita dapatkan juga bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack Q P with rightwards arrow on top close vertical bar end cell equals cell square root of 2 squared plus open parentheses negative 3 close parentheses squared plus 5 squared end root end cell row blank equals cell square root of 4 plus 9 plus 25 end root end cell row blank equals cell square root of 38 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack Q R with rightwards arrow on top close vertical bar end cell equals cell square root of 3 squared plus 1 squared plus 2 squared end root end cell row blank equals cell square root of 9 plus 1 plus 4 end root end cell row blank equals cell square root of 14 end cell row blank blank blank end table end style

Oleh karena itu, kita dapatkan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator stack Q P with rightwards arrow on top times stack Q R with rightwards arrow on top over denominator open vertical bar stack Q P with rightwards arrow on top close vertical bar open vertical bar stack Q R with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 13 over denominator square root of 38 times square root of 14 end fraction end cell row blank equals cell fraction numerator 13 over denominator square root of 532 end fraction end cell row blank blank blank end table end style

Perhatikan bahwa cos α bernilai positif.
Sehingga α berada di kuadran I atau IV.
Karena sudut yang dibentuk vektor berkisar di kuadran I atau II, maka haruslah α berada di kuadran I.

Perhatikan gambar berikut!

Dapat diperhatikan bahwa begin mathsize 14px style cos space alpha equals fraction numerator 13 over denominator square root of 532 end fraction end style

Karena segitiga di atas merupakan segitiga siku-siku, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus 13 squared end cell equals cell open parentheses square root of 532 close parentheses squared end cell row cell x squared plus 169 end cell equals 532 row cell x squared end cell equals 363 row x equals cell plus-or-minus square root of 363 end cell end table end style

Karena x menyatakan panjang sebuah sisi segitiga, maka x tidak mungkin bernilai negatif. Oleh karena itu, begin mathsize 14px style x equals square root of 363 end style
Jadi, didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application alpha end cell equals cell fraction numerator x over denominator square root of 532 end fraction end cell row blank equals cell fraction numerator square root of 363 over denominator square root of 532 end fraction end cell row blank equals cell fraction numerator 11 square root of 3 over denominator 2 square root of 133 end fraction end cell row blank equals cell fraction numerator 11 square root of 3 over denominator 2 square root of 133 end fraction times fraction numerator square root of 133 over denominator square root of 133 end fraction end cell row blank equals cell fraction numerator 11 square root of 399 over denominator 2 times 133 end fraction end cell row blank equals cell fraction numerator 11 square root of 399 over denominator 266 end fraction end cell row blank blank blank end table end style

Jadi, jawaban yang tepat adalah D.

 

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