Tentukan nilai x yang memenuhi persamaan: sin(22016x)1=220162cos (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 $0\leq x\leq\mathrm\pi$ .

S. Difhayanti

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Jawaban terverifikasi

Jawaban

nilai

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:

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