Pertanyaan

Buktikan cos 2 x + cos 4 x + cos 6 x sin 2 x + sin 4 x + sin 6 x = tan x .

Buktikan $\frac{sin 2 x + sin 4 x + sin 6 x}{cos 2 x + cos 4 x + cos 6 x} = tan x$ .

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Klaim

N. Puspita

Master Teacher

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Pembahasan

Gunakan konsep rumus penjumlahan trigonometri sinus dan cosinus. Akan dibuktikan . Perhatikan perhitungan berikut. Tidak tebukti bahwa , karena yang benar adalah . Jadi, tidak terbukti.

Gunakan konsep rumus penjumlahan trigonometri sinus dan cosinus.

$table attributes columnalign right center left columnspacing 2px end attributes row cell sin space alpha plus sin space beta end cell equals cell 2 space sin space 1 half open parentheses alpha plus beta close parentheses space cos space 1 half open parentheses alpha minus beta close parentheses end cell row cell cos space alpha plus cos space beta end cell equals cell 2 space cos space 1 half open parentheses alpha plus beta close parentheses space cos space 1 half open parentheses alpha minus beta close parentheses end cell row cell tan space alpha end cell equals cell fraction numerator sin space alpha over denominator cos space alpha end fraction end cell end table$

Akan dibuktikan $fraction numerator sin space 2 x plus sin space 4 x plus sin space 6 x over denominator cos space 2 x plus cos space 4 x plus cos space 6 x end fraction equals tan space x$ .

Perhatikan perhitungan berikut.

$table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell fraction numerator sin space 2 x plus sin space 4 x plus sin space 6 x over denominator cos space 2 x plus cos space 4 x plus cos space 6 x end fraction end cell row blank equals cell fraction numerator open parentheses sin space 6 x plus sin space 2 x close parentheses plus sin space 4 x over denominator open parentheses cos space 6 x plus cos space 2 x close parentheses plus cos space 4 x end fraction end cell row blank equals cell fraction numerator open parentheses 2 space sin space begin display style 1 half end style open parentheses 6 x plus 2 x close parentheses space cos space begin display style 1 half end style open parentheses 6 x minus 2 x close parentheses close parentheses plus sin space 4 x over denominator open parentheses 2 space cos space begin display style 1 half end style open parentheses 6 x plus 2 x close parentheses space cos space begin display style 1 half end style open parentheses 6 x minus 2 x close parentheses close parentheses plus cos space 4 x end fraction end cell row blank equals cell fraction numerator open parentheses 2 space sin space begin display style 1 half end style open parentheses 8 x close parentheses space cos space begin display style 1 half end style open parentheses 4 x close parentheses close parentheses plus sin space 4 x over denominator open parentheses 2 space cos space begin display style 1 half end style open parentheses 8 x close parentheses space cos space begin display style 1 half end style open parentheses 4 x close parentheses close parentheses plus cos space 4 x end fraction end cell row blank equals cell fraction numerator open parentheses 2 space sin space 4 x space cos space 2 x close parentheses plus sin space 4 x over denominator open parentheses 2 space cos space 4 x space cos space 2 x close parentheses plus cos space 4 x end fraction end cell row blank equals cell fraction numerator 2 space sin space 4 x space cos space 2 x plus sin space 4 x over denominator 2 space cos space 4 x space cos space 2 x plus cos space 4 x end fraction end cell row blank equals cell fraction numerator sin space 4 x up diagonal strike open parentheses 2 space cos space 2 x plus 1 close parentheses end strike over denominator cos space 4 x up diagonal strike open parentheses 2 space cos space 2 x plus 1 close parentheses end strike end fraction end cell row blank equals cell fraction numerator sin space 4 x over denominator cos space 4 x end fraction end cell row blank equals cell tan space 4 x end cell end table$

Tidak tebukti bahwa $fraction numerator sin space 2 x plus sin space 4 x plus sin space 6 x over denominator cos space 2 x plus cos space 4 x plus cos space 6 x end fraction equals tan space x$ , karena yang benar adalah $fraction numerator sin space 2 x plus sin space 4 x plus sin space 6 x over denominator cos space 2 x plus cos space 4 x plus cos space 6 x end fraction equals tan space 4 x$ .

Jadi, tidak terbukti.