Pertanyaan

Tentukan ekspresi aljabar dari setiap ekspresi berikut. c. sin ( cos − 1 x − sin − 1 3 x )

Tentukan ekspresi aljabar dari setiap ekspresi berikut.

c. $sin (cos^{- 1} x - sin^{- 1} 3 x)$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

.

sin space open parentheses cos to the power of negative 1 end exponent space x minus sin to the power of negative 1 end exponent space 3 x close parentheses equals open parentheses square root of 1 minus x squared end root close parentheses open parentheses square root of 1 minus 9 x squared end root close parentheses minus 3 x squared

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 sin space open parentheses cos to the power of negative 1 end exponent space x minus sin to the power of negative 1 end exponent space 3 x close parentheses . Misalkan:

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

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

- Untuk sin to the power of negative 1 end exponent space 3 x

$table attributes columnalign right center left columnspacing 0px end attributes row beta equals cell sin to the power of negative 1 end exponent space 3 x end cell row cell sin space beta end cell equals cell 3 x end cell row cell sin space beta end cell equals cell fraction numerator 3 x over denominator 1 end fraction space rightwards arrow space fraction numerator sisi space depan over denominator sisi space miring end fraction end cell end table$

Perhatikan gambar berikut:

Maka:

p equals square root of 1 squared minus x squared end root p equals square root of 1 minus x squared end root dan q equals square root of 1 squared minus open parentheses 3 x close parentheses squared end root q equals square root of 1 minus 9 x squared end root

Sehingga untuk sin space open parentheses x minus y close parentheses equals sin space x space cos space y minus cos space x space sin space y , digunakan sebagai berikut:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses cos to the power of negative 1 end exponent space x minus sin to the power of negative 1 end exponent space 3 x close parentheses end cell equals cell sin space open parentheses theta minus beta close parentheses end cell row blank equals cell sin space theta space cos space beta minus cos space theta space sin space beta end cell row blank equals cell fraction numerator sisi space depan space theta over denominator sisi space miring space theta end fraction times fraction numerator sisi space samping space beta over denominator sisi space miring space beta end fraction minus fraction numerator sisi space samping space theta over denominator sisi space miring space theta end fraction times fraction numerator sisi space depan space beta over denominator sisi space miring space beta end fraction end cell row blank equals cell fraction numerator square root of 1 minus x squared end root over denominator 1 end fraction times fraction numerator square root of 1 minus 9 x squared end root over denominator 1 end fraction minus x over 1 times fraction numerator 3 x over denominator 1 end fraction end cell row blank equals cell open parentheses square root of 1 minus x squared end root close parentheses open parentheses square root of 1 minus 9 x squared end root close parentheses minus 3 x squared end cell end table end style$

Jadi, sin space open parentheses cos to the power of negative 1 end exponent space x minus sin to the power of negative 1 end exponent space 3 x close parentheses equals open parentheses square root of 1 minus x squared end root close parentheses open parentheses square root of 1 minus 9 x squared end root close parentheses minus 3 x squared .