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Himpunan penyelesaian dari pertidaksamaan sin squared 2 x times open parentheses cotan squared x plus 1 close parentheses plus square root of 2 cos invisible function application x greater than 0 pada interval fraction numerator blank pi over denominator 2 end fraction less than x less than fraction numerator 3 pi over denominator 2 end fraction adalah ….

  1. fraction numerator blank 5 pi over denominator 6 end fraction less than x less than fraction numerator blank 7 pi over denominator 6 end fraction 

  2. pi over 2 less than x less than fraction numerator blank 3 pi over denominator 4 end fraction 

  3. fraction numerator blank 3 pi over denominator 4 end fraction less than x less than fraction numerator 5 pi over denominator 4 end fraction 

  4. x less than fraction numerator blank 3 pi over denominator 4 end fraction atau x greater than fraction numerator 5 pi over denominator 4 end fraction 

  5. x less than fraction numerator blank 5 pi over denominator 6 end fraction atau x greater than fraction numerator blank 7 pi over denominator 6 end fraction 

R. Tri

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Ingat identitas trigonometri cotan squared x plus 1 equals cosec squared x.

Ingat pula bahwa sin invisible function application 2 x equals 2 sin invisible function application x cos invisible function application x.

Perhatikan perhitungan berikut ini!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared 2 x times open parentheses cotan squared x plus 1 close parentheses plus square root of 2 cos invisible function application x end cell greater than 0 row cell 2 sin squared x cos squared x times cosec squared x plus square root of 2 cos invisible function application x end cell greater than 0 row cell 2 sin squared x cos squared x times fraction numerator 1 over denominator sin squared x end fraction plus square root of 2 cos invisible function application x end cell greater than 0 row cell 2 cos squared x plus square root of 2 cos invisible function application x end cell greater than 0 row cell cos invisible function application x open parentheses 2 cos invisible function application x plus square root of 2 close parentheses end cell greater than 0 end table

Kemudian, tentukan pembuat nol serta nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos invisible function application x end cell equals 0 row x equals cell open curly brackets fraction numerator blank pi over denominator 2 end fraction comma blank fraction numerator 3 pi over denominator 2 end fraction close curly brackets end cell end table

dan

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 cos invisible function application x plus square root of 2 end cell equals 0 row cell 2 cos invisible function application x end cell equals cell negative square root of 2 end cell row cell cos invisible function application x end cell equals cell negative fraction numerator square root of 2 over denominator 2 end fraction end cell end table

Ingat nilai cos invisible function application 45 degree equals fraction numerator square root of 2 over denominator 2 end fraction. Ingat pula bahwa cosinus bernilai negatif pada kuadran II dan III. 

Akibatnya nilai x yang memenuhi adalah sebagai berikut.

cos invisible function application x equals cos invisible function application left parenthesis 180 minus 45 right parenthesis degree equals cos invisible function application 135 degree x equals 135 degree equals fraction numerator 3 pi over denominator 4 end fraction

dan

cos invisible function application x equals cos invisible function application open parentheses 180 plus 45 close parentheses degree equals cos invisible function application 225 degree x equals 225 degree equals fraction numerator 5 pi over denominator 4 end fraction

Didapat nilai x equals open curly brackets fraction numerator blank pi over denominator 2 end fraction comma fraction numerator 3 pi over denominator 4 end fraction comma fraction numerator 5 pi over denominator 4 end fraction comma blank fraction numerator 3 pi over denominator 2 end fraction close curly brackets

Selanjutnya, setelah dilakukan uji titik didapat garis bilangan sebagai berikut.
 


Sebelumnya, telah didapat 2 cos squared x plus cos invisible function application x greater than 0 dengan interval fraction numerator blank pi over denominator 2 end fraction less than x less than fraction numerator 3 pi over denominator 2 end fraction.

Dengan demikian, pilihlah daerah bernilai positif yang memenuhi interval tersebut sehingga didapat nilai x yang memenuhi adalah fraction numerator blank 3 pi over denominator 4 end fraction less than x less than fraction numerator 5 pi over denominator 4 end fraction.

Jadi, jawaban yang tepat adalah C.

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