Pertanyaan

Jika y = tan x , tunjukkan bahwa: a. sin 2 x = 1 + y 2 2 y b. cos 2 x = 1 + y 2 1 − y 2 c. tan 2 x = 1 − y 2 2 y

Jika $y = tan x$ , tunjukkan bahwa:

a. $sin 2 x = \frac{2 y}{1 + y ^{2}}$

b. $cos 2 x = \frac{1 - y ^{2}}{1 + y ^{2}}$

c. $tan 2 x = \frac{2 y}{1 - y ^{2}}$

O. Rahmawati

Master Teacher

Mahasiswa/Alumni UIN Sunan Gunung Djati Bandung

Jawaban terverifikasi

Pembahasan

a. Rumus sudut rangkap sinus: Terbukti bahwa . b. Rumus sudut rangkap cosinus: Terbukti bahwa . c. Rumus sudut rangkap tangen: Terbukti bahwa .

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space x space end cell equals cell y over 1 equals y over x end cell row r equals cell square root of x squared plus y squared end root end cell row r equals cell square root of 1 squared plus y squared end root end cell end table

a. Rumus sudut rangkap sinus:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 x space end cell equals cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell row cell 2 space sin space x space cos space x end cell equals cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell row cell 2 open parentheses y over r close parentheses open parentheses x over r close parentheses end cell equals cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell row cell 2 open parentheses fraction numerator y over denominator square root of 1 plus y squared end root end fraction close parentheses open parentheses fraction numerator 1 over denominator square root of 1 plus y squared end root end fraction close parentheses end cell equals cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell row cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell equals cell fraction numerator 2 y over denominator 1 plus y squared end fraction end cell end table$

Terbukti bahwa $sin space 2 x equals fraction numerator 2 y over denominator 1 plus y squared end fraction$ .

b. Rumus sudut rangkap cosinus:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 x end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell row cell cos squared x minus sin squared x end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell row cell fraction numerator cos squared x minus sin squared x over denominator 1 end fraction end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell row cell fraction numerator begin display style fraction numerator cos squared x minus sin squared x over denominator cos squared x end fraction end style over denominator begin display style fraction numerator cos squared x plus sin squared x over denominator cos squared x end fraction end style end fraction end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell row cell fraction numerator 1 minus tan squared x over denominator 1 plus tan squared x end fraction end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell row cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell equals cell fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction end cell end table$

Terbukti bahwa $cos space 2 x equals fraction numerator 1 minus y squared over denominator 1 plus y squared end fraction$ .

c. Rumus sudut rangkap tangen:

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space 2 x end cell equals cell fraction numerator 2 y over denominator 1 minus y squared end fraction end cell row cell fraction numerator 2 space tan space x over denominator 1 minus tan squared x end fraction end cell equals cell fraction numerator 2 y over denominator 1 minus y squared end fraction end cell row cell fraction numerator 2 y over denominator 1 minus y squared end fraction end cell equals cell fraction numerator 2 y over denominator 1 minus y squared end fraction end cell end table$

Terbukti bahwa $table attributes columnalign right center left columnspacing 0px end attributes row blank blank tan end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 y over denominator 1 minus y squared end fraction end cell end table$ .

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