Pertanyaan

Buktikan identitas berikut. c. 1 + sin 2 θ + cos 2 θ 1 + sin 2 θ − cos 2 θ = tan θ

Buktikan identitas berikut.

c. $\frac{1 + sin 2 θ - cos 2 θ}{1 + sin 2 θ + cos 2 θ} = tan θ$ $\;$

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A. Armanda

Master Teacher

Mahasiswa/Alumni Universitas Indraprasta PGRI

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa $fraction numerator 1 plus sin space 2 theta minus cos space 2 theta over denominator 1 plus sin space 2 theta plus cos space 2 theta end fraction equals tan space theta$ .

Pembahasan

Ingat! Sudut Rangkap pada Sinus Sudut Rangkap pada Cosinus Akan dibuktikan bahwa . Bukti: Dengan menggunakan rumus sudut rangkap pada sinus dan cosinus, maka: Terbukti! Jadi, terbukti bahwa .

Ingat!

$tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction$

Sudut Rangkap pada Sinus

Sudut Rangkap pada Cosinus

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 alpha end cell equals cell cos squared space alpha minus sin squared space alpha end cell row blank equals cell 2 space cos squared space alpha minus 1 end cell row blank equals cell 1 minus 2 space sin squared space alpha end cell end table

Akan dibuktikan bahwa $fraction numerator 1 plus sin space 2 theta minus cos space 2 theta over denominator 1 plus sin space 2 theta plus cos space 2 theta end fraction equals tan space theta$ .

Bukti:

Dengan menggunakan rumus sudut rangkap pada sinus dan cosinus, maka:

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 plus sin space 2 theta minus cos space 2 theta over denominator 1 plus sin space 2 theta plus cos space 2 theta end fraction end cell row blank equals cell fraction numerator 1 plus 2 space sin space theta space cos space theta minus open parentheses 1 minus 2 space sin squared space theta close parentheses over denominator 1 plus 2 space sin space theta space cos space theta plus open parentheses 2 space cos squared space theta minus 1 close parentheses end fraction end cell row blank equals cell fraction numerator down diagonal strike 1 plus 2 space sin space theta space cos space theta down diagonal strike negative 1 end strike plus 2 space sin squared space theta over denominator up diagonal strike 1 plus 2 space sin space theta space cos space theta plus 2 space cos squared space theta up diagonal strike negative 1 end strike end fraction end cell row blank equals cell fraction numerator 2 space sin space theta space cos space theta plus 2 space sin squared space theta over denominator 2 space sin space theta space cos space theta plus 2 space cos squared space theta end fraction end cell row blank equals cell fraction numerator down diagonal strike 2 space sin space theta space up diagonal strike open parentheses cos space theta plus sin space theta close parentheses end strike over denominator down diagonal strike 2 space cos space theta space up diagonal strike open parentheses sin space theta plus cos space theta close parentheses end strike end fraction end cell row blank equals cell fraction numerator sin space theta over denominator cos space theta end fraction end cell row blank equals cell tan space theta end cell end table$

Terbukti!

Jadi, terbukti bahwa $fraction numerator 1 plus sin space 2 theta minus cos space 2 theta over denominator 1 plus sin space 2 theta plus cos space 2 theta end fraction equals tan space theta$ .