Buktikanlah setiap identitas berikut. cos 3α−cos αsin 3α−sin α=−cotan 2α

Pertanyaan

Buktikanlah setiap identitas berikut.

$fraction numerator sin space 3 alpha minus sin space alpha over denominator cos space 3 alpha minus cos space alpha end fraction equals negative cotan space space 2 alpha$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

terbukti bahwa $fraction numerator sin space 3 alpha minus sin space alpha over denominator cos space 3 alpha minus cos space alpha end fraction equals negative cotan space space 2 alpha$ .

Pembahasan

Ingat rumus jumlah dan selisih trigonometri berikut ini:

$\sin\;A-\sin\;B=2\;\cos\;\frac12\left(A+B\right)\;\sin\;\frac12\left(A-B\right)$

$\cos\;A-\cos\;B=-2\;\sin\;\frac12\left(A+B\right)\;\sin\;\frac12\left(A-B\right)$

Dengan menggunakan konsep di atas, diperoleh hasil:

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space 3 alpha minus sin space alpha over denominator cos space space 3 alpha minus cos space alpha end fraction end cell equals cell fraction numerator 2 space cos space 1 half open parentheses 3 alpha plus alpha close parentheses space sin space 1 half open parentheses 3 alpha minus alpha close parentheses over denominator negative 2 space sin space 1 half open parentheses 3 alpha plus alpha close parentheses space sin space 1 half open parentheses 3 alpha minus alpha close parentheses end fraction end cell row blank equals cell negative fraction numerator cos space 1 half open parentheses 4 alpha close parentheses space sin space 1 half open parentheses 2 alpha close parentheses over denominator sin space 1 half open parentheses 4 alpha close parentheses space sin space 1 half open parentheses 2 alpha close parentheses end fraction end cell row blank equals cell negative fraction numerator cos space 2 alpha space sin space alpha over denominator sin space 2 alpha space sin space alpha end fraction end cell row blank equals cell negative fraction numerator cos space 2 alpha over denominator sin space 2 alpha end fraction end cell row blank equals cell negative cotan space 2 straight alpha end cell end table$

Jadi, terbukti bahwa $fraction numerator sin space 3 alpha minus sin space alpha over denominator cos space 3 alpha minus cos space alpha end fraction equals negative cotan space space 2 alpha$ .