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Tentukan nilai x yang memenuhi persamaan: sin(22016x​)1​=220162​cos (2x​)⋅cos(4x​)⋅...⋅cos (22016x​) untuk 0≤x≤π.

Pertanyaan

Tentukan nilai x yang memenuhi persamaan:

fraction numerator 1 over denominator sin open parentheses begin display style x over 2 to the power of 2016 end style close parentheses end fraction equals 2 to the power of 2016 square root of 2 cos space open parentheses x over 2 close parentheses times cos open parentheses x over 4 close parentheses times... times cos space open parentheses x over 2 to the power of 2016 close parentheses

untuk 0 less or equal than x less or equal than straight pi.

S. Difhayanti

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Jawaban terverifikasi

Jawaban

nilai x yang memenuhi adalah x equals straight pi over 4 atau space x equals fraction numerator 3 straight pi over denominator 4 end fraction.

Pembahasan

Ingat bahwa:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank alpha end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank alpha end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank alpha end table

Sehingga, didapat perhitungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space alpha space cos space alpha end cell equals cell 1 half sin space 2 alpha end cell row cell fraction numerator 1 over denominator sin open parentheses begin display style x over 2 to the power of 2016 end style close parentheses end fraction end cell equals cell 2 to the power of 2016 square root of 2 cos space open parentheses x over 2 close parentheses times cos open parentheses x over 4 close parentheses times... times cos space open parentheses x over 2 to the power of 2016 close parentheses end cell row 1 equals cell 2 to the power of 2016 square root of 2 cos space open parentheses x over 2 close parentheses times cos open parentheses x over 4 close parentheses times... times cos space open parentheses x over 2 to the power of 2016 close parentheses space sin space open parentheses x over 2 to the power of 2016 close parentheses end cell row 1 equals cell 2 to the power of 2015 square root of 2 cos space open parentheses x over 2 close parentheses times cos open parentheses x over 4 close parentheses times... times sin space open parentheses x over 2 to the power of 2015 close parentheses end cell row 1 equals cell 2 to the power of 2014 square root of 2 cos space open parentheses x over 2 close parentheses times cos open parentheses x over 4 close parentheses times... times sin space open parentheses x over 2 to the power of 2014 close parentheses end cell row 1 equals cell square root of 2 sin space x end cell row cell sin space x end cell equals cell straight pi over 4 end cell end table

Maka, x equals straight pi over 4 atau space x equals fraction numerator 3 straight pi over denominator 4 end fraction.

Jadi, nilai x yang memenuhi adalah x equals straight pi over 4 atau space x equals fraction numerator 3 straight pi over denominator 4 end fraction.

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