Pertanyaan

Tentukan ekspresi aljabar dari setiap ekspresi berikut. d. tan ( sin − 1 2 x + cos − 1 ( x 1 ) )

Tentukan ekspresi aljabar dari setiap ekspresi berikut.

d. $tan (sin^{- 1} 2 x + cos^{- 1} (\frac{1}{x}))$

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D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

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$tan space open parentheses sin to the power of negative 1 end exponent space 2 x plus cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x plus square root of x squared minus 1 end root times square root of 1 minus 4 x squared end root over denominator square root of 1 minus 4 x squared minus end root 2 x square root of x squared minus 1 end root end fraction end cell end table$ .

Pembahasan

Ingat perbandingan sisi trigonometri berikut: Diketahui ekspresi . Misalkan: - Untuk - Untuk Perhatikan gambar berikut: Maka: dan Sehingga untuk ,digunakan sebagai berikut: Jadi, .

Ingat perbandingan sisi trigonometri berikut:

$sin space theta equals fraction numerator sisi space depan over denominator sisi space miring end fraction rightwards arrow cosec space theta equals fraction numerator sisi space miring over denominator sisi space depan end fraction cos space theta equals fraction numerator sisi space samping over denominator sisi space miring end fraction rightwards arrow sec space theta equals fraction numerator sisi space miring over denominator sisi space depan end fraction tan space theta equals fraction numerator sisi space depan over denominator sisi space samping end fraction rightwards arrow cotan space theta equals fraction numerator sisi space miring over denominator sisi space depan end fraction$

Diketahui ekspresi tan space open parentheses sin to the power of negative 1 end exponent space 2 x plus cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses close parentheses . Misalkan:

- Untuk sin to the power of negative 1 end exponent space 2 x

$table attributes columnalign right center left columnspacing 0px end attributes row theta equals cell sin to the power of negative 1 end exponent space 2 x end cell row cell sin space theta end cell equals cell 2 x end cell row cell sin space theta end cell equals cell fraction numerator 2 x over denominator 1 end fraction space rightwards arrow fraction numerator sisi space depan over denominator sisi space miring end fraction end cell end table$

- Untuk cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses

$table attributes columnalign right center left columnspacing 0px end attributes row beta equals cell cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses end cell row cell cos space beta end cell equals cell 1 over x space rightwards arrow space fraction numerator sisi space samping over denominator sisi space miring end fraction end cell end table$

Perhatikan gambar berikut:

Maka:

p equals square root of 1 squared minus open parentheses 2 x close parentheses squared end root p equals square root of 1 minus 4 x squared end root dan q equals square root of x squared minus 1 squared end root q equals square root of x squared minus 1 end root

Sehingga untuk $tan space open parentheses x plus y close parentheses equals fraction numerator tan space x plus tan space y over denominator 1 minus tan space x space tan space y end fraction$ , digunakan sebagai berikut:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan space open parentheses sin to the power of negative 1 end exponent space 2 x plus cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses close parentheses end cell equals cell tan space open parentheses theta plus beta close parentheses end cell row blank equals cell fraction numerator tan space theta plus tan space beta over denominator 1 minus tan space theta space tan space beta end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x over denominator square root of 1 minus 4 x squared end root end fraction end style plus begin display style fraction numerator square root of x squared minus 1 end root over denominator 1 end fraction end style over denominator 1 minus begin display style fraction numerator 2 x over denominator square root of 1 minus 4 x squared end root end fraction end style times begin display style fraction numerator square root of x squared minus 1 end root over denominator 1 end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x plus square root of x squared minus 1 end root times square root of 1 minus 4 x squared end root over denominator square root of 1 minus 4 x squared end root end fraction end style over denominator 1 minus begin display style fraction numerator 2 x square root of x squared minus 1 end root over denominator square root of 1 minus 4 x squared end root end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x plus square root of x squared minus 1 end root times square root of 1 minus 4 x squared end root over denominator up diagonal strike square root of 1 minus 4 x squared end root end strike end fraction end style over denominator begin display style fraction numerator square root of 1 minus 4 x squared minus end root 2 x square root of x squared minus 1 end root over denominator up diagonal strike square root of 1 minus 4 x squared end root end strike end fraction end style end fraction end cell row blank equals cell fraction numerator 2 x plus square root of x squared minus 1 end root times square root of 1 minus 4 x squared end root over denominator square root of 1 minus 4 x squared minus end root 2 x square root of x squared minus 1 end root end fraction end cell end table end style$

Jadi, $tan space open parentheses sin to the power of negative 1 end exponent space 2 x plus cos to the power of negative 1 end exponent space open parentheses 1 over x close parentheses close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x plus square root of x squared minus 1 end root times square root of 1 minus 4 x squared end root over denominator square root of 1 minus 4 x squared minus end root 2 x square root of x squared minus 1 end root end fraction end cell end table$ .

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