Pertanyaan

Jika sin ( α + β ) cos γ = 2 sin α cos ( β − γ ) , buktikanlah : tan γ = 2 sin α sin β sin ( β − α ) .

Jika $sin (α + β) cos γ = 2 sin α cos (β - γ)$ , buktikanlah : $tan γ = \frac{sin ( β - α )}{2 sin α sin β}$ .

H. Janatu

Master Teacher

Mahasiswa/Alumni Universitas Riau

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa $tan space gamma equals fraction numerator sin space left parenthesis beta minus alpha right parenthesis over denominator 2 space sin space alpha space sin space beta end fraction$ .

Pembahasan

Ingat rumus penjumlahan sudut pada sinus dan cosinus, Berdasarkan rumus tersebut, maka Dengan demikian terbukti bahwa .

Ingat rumus penjumlahan sudut pada sinus dan cosinus,

Berdasarkan rumus tersebut, maka

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis alpha plus beta right parenthesis space cos space gamma end cell equals cell 2 space sin space alpha space cos space left parenthesis beta minus gamma right parenthesis end cell row cell open parentheses sin space alpha space cos space beta plus cos space alpha space sin space beta close parentheses cos space gamma end cell equals cell 2 space sin space alpha space left parenthesis cos space beta space cos space gamma plus sin space beta space sin space gamma right parenthesis end cell row cell cos space alpha space sin space beta space cos space gamma minus sin space alpha space cos space beta space cos space gamma end cell equals cell 2 space sin space alpha space sin space beta space sin space gamma end cell row cell open parentheses cos space alpha space sin space beta minus sin space alpha space cos space beta close parentheses cos space gamma end cell equals cell 2 space sin space alpha space sin space beta space sin space gamma end cell row cell sin space left parenthesis beta minus alpha right parenthesis space cos space gamma end cell equals cell 2 space sin space alpha space sin space beta space sin space gamma end cell row cell fraction numerator sin space left parenthesis beta minus alpha right parenthesis over denominator 2 space sin space alpha space sin space beta end fraction end cell equals cell fraction numerator space sin space gamma over denominator cos space gamma end fraction end cell row cell fraction numerator sin space left parenthesis beta minus alpha right parenthesis over denominator 2 space sin space alpha space sin space beta end fraction end cell equals cell tan space gamma end cell end table end style$

Dengan demikian terbukti bahwa $tan space gamma equals fraction numerator sin space left parenthesis beta minus alpha right parenthesis over denominator 2 space sin space alpha space sin space beta end fraction$ .