Pertanyaan

Dengan menggunakan formula trigonometri sin ( A + B ) = sin A cos B + cos A sin B , tentukan ekspresi aljabar untuk ekspresi sin [ tan − 1 ( x ) + cos − 1 ( 2 x ) ] .

Dengan menggunakan formula trigonometri $sin (A + B) = sin A cos B + cos A sin B$ , tentukan ekspresi aljabar untuk ekspresi $sin [tan^{- 1} (x) + cos^{- 1} (2 x)]$ .

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

.

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

Pembahasan

Ingat perbandingan sisi trigonometri berikut: Diketahui ekspresi . Misalkan: dan Perhatikan gambar berikut: Maka: dan Sehingga formula , dapat 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 square brackets tan to the power of negative 1 end exponent space open parentheses x close parentheses plus cos to the power of negative 1 end exponent space open parentheses 2 x close parentheses close square brackets . Misalkan:

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

dan

$table attributes columnalign right center left columnspacing 0px end attributes row beta equals cell cos to the power of negative 1 end exponent space open parentheses 2 x close parentheses end cell row cell tan space beta end cell equals cell 2 x end cell row cell tan space beta end cell equals cell fraction numerator 2 x over denominator 1 end fraction end cell row cell tan space beta end cell equals cell fraction numerator 2 x over denominator 1 end fraction rightwards arrow fraction numerator sisi space depan over denominator sisi space samping end fraction end cell end table$

Perhatikan gambar berikut:

Maka:

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

Sehingga formula sin space open parentheses A plus B close parentheses equals sin space A space cos space B plus cos space A space sin space B , dapat digunakan sebagai berikut:

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

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