Pertanyaan

Tentukan himpunan penyelesaian dari setiap persamaan trigonometri berikut. b. cos ( 4 3 x − π ) = sin ( 4 π − 2 x ) , 0 ≤ x ≤ π

Tentukan himpunan penyelesaian dari setiap persamaan trigonometri berikut.

b. $cos (\frac{3}{4} x - π) = sin (\frac{π}{4} - 2 x), 0 \leq x \leq π$

A. Khairunisa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Jawaban

himpunan penyelesaian dari persamaan tersebut adalah .

himpunan penyelesaian dari persamaan tersebut adalah HP equals open curly brackets 3 over 11 straight pi comma space straight pi close curly brackets

HP equals open curly brackets 3 over 11 straight pi comma space straight pi close curly brackets

Pembahasan

Samakan kedua ruas menjadi sinus, lalu tentukan nilai x.Penyelesaian dari persamaan trigonometri yaitu atau , dengan k anggota bilangan bulat. atau Dengan demikian, himpunan penyelesaian dari persamaan tersebut adalah .

Samakan kedua ruas menjadi sinus, lalu tentukan nilai x. Penyelesaian dari persamaan trigonometri cos space x equals cos space alpha degree yaitu x equals straight alpha plus 2 πk atau straight x equals open parentheses straight pi minus straight alpha close parentheses plus 2 πk , dengan k anggota bilangan bulat.

$cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses straight pi over 4 minus 2 straight x close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals cos space open parentheses straight pi over 2 minus open parentheses straight pi over 4 minus 2 straight x close parentheses close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses fraction numerator 2 straight pi over denominator 4 end fraction minus straight pi over 4 plus 2 straight x close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses straight pi over 4 plus 2 straight x close parentheses space space space space space space space space space space 3 over 4 straight x minus straight pi equals straight pi over 4 plus 2 straight x space plus open parentheses 2 πk close parentheses space space space space space space space space space 3 over 4 straight x minus 2 straight x equals straight pi over 4 plus straight pi plus open parentheses 2 πk close parentheses space space space space space space space space 3 over 4 straight x minus 8 over 4 straight x equals straight pi over 4 plus fraction numerator 4 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses space space space space space space space space space space space space space space minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses untuk space straight k equals 0 minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus open parentheses 2 straight pi times 0 close parentheses minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction space space space space space space space straight x equals negative straight pi space rightwards arrow tidak space memenuhi untuk space straight k equals 1 minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus open parentheses 2 straight pi times 1 close parentheses minus 5 over 4 straight x equals fraction numerator 5 straight pi over denominator 4 end fraction plus 2 straight pi space minus 5 straight x equals 13 straight pi space space space space space space space straight x equals negative 13 over 5 straight pi space rightwards arrow tidak space memenuhi$

atau

$cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses straight pi over 4 minus 2 straight x close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals cos space open parentheses straight pi over 2 minus open parentheses straight pi over 4 minus 2 straight x close parentheses close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses fraction numerator 2 straight pi over denominator 4 end fraction minus straight pi over 4 plus 2 straight x close parentheses cos space open parentheses 3 over 4 straight x minus straight pi close parentheses equals sin space open parentheses straight pi over 4 plus 2 straight x close parentheses space space space space space space space space space space 3 over 4 straight x minus straight pi equals negative open parentheses straight pi over 4 plus 2 straight x close parentheses space plus open parentheses 2 πk close parentheses space space space space space space space space space 3 over 4 straight x plus 2 straight x equals negative straight pi over 4 plus straight pi plus open parentheses 2 πk close parentheses space space space space space space space space 3 over 4 straight x plus 8 over 4 straight x equals negative straight pi over 4 plus fraction numerator 4 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses space space space space space space space space space space space space space space space 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses untuk space straight k equals 0 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus open parentheses 2 straight pi times 0 close parentheses space space space space space space straight x equals fraction numerator 3 straight pi over denominator 11 end fraction untuk space straight k equals 1 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus open parentheses 2 πk close parentheses 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus open parentheses 2 straight pi times 1 close parentheses 11 over 4 straight x equals fraction numerator 3 straight pi over denominator 4 end fraction plus 2 straight pi 11 over 4 straight x equals fraction numerator 11 straight pi over denominator 4 end fraction space space space space space straight x equals straight pi$

Dengan demikian, himpunan penyelesaian dari persamaan tersebut adalah HP equals open curly brackets 3 over 11 straight pi comma space straight pi close curly brackets .