Pertanyaan

Jika sin θ = 2 3 x dan 0 < θ < 2 π , nyatakan ekspresi 4 θ − 2 sin θ ⋅ cos θ dalam fungsi dari x

Jika $sin θ = \frac{3 x}{2}$ dan $0 < θ < \frac{π}{2}$ , nyatakan ekspresi $\frac{θ}{4} - 2 sin θ \cdot cos θ$ dalam fungsi dari $x$

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Klaim

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

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$theta over 4 minus 2 space sin space theta times cos space theta equals fraction numerator sin to the power of negative 1 end exponent space open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator 4 end fraction minus fraction numerator 3 x square root of 4 minus 9 x squared end root over denominator 2 end fraction$ .

Pembahasan

Ingat perbandingan sisi trigonometri berikut: Jika dan , maka: dan Perhatikan gambar berikut: tentukan nilai terlebih dahulu dengan cara: Interval berada pada kuadran I, berarti seluruh nilai perbandingan trigonometri bernilai positif. Sehingga ekspresi dalam fungsi dari dapat ditentukan dengan cara: Jadi, .

Ingat perbandingan sisi trigonometri berikut:

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

Jika $sin space theta equals fraction numerator 3 x over denominator 2 end fraction$ dan 0 less than theta less than straight pi over 2 , maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space theta end cell equals cell fraction numerator 3 x over denominator 2 end fraction end cell row theta equals cell sin to the power of negative 1 end exponent space open parentheses fraction numerator 3 x over denominator 2 end fraction close parentheses end cell end table$

dan

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space theta end cell equals cell fraction numerator 3 x over denominator 2 end fraction end cell row cell fraction numerator sisi space depan over denominator sisi space miring end fraction end cell equals cell fraction numerator 3 x over denominator 2 end fraction end cell end table$

Perhatikan gambar berikut:

tentukan nilai terlebih dahulu dengan cara:

p equals square root of 2 squared minus open parentheses 3 x close parentheses squared end root p equals square root of 4 minus 9 x squared end root

Interval 0 less than theta less than straight pi over 2 berada pada kuadran I, berarti seluruh nilai perbandingan trigonometri bernilai positif. Sehingga ekspresi theta over 4 minus 2 space sin space theta times cos space theta dalam fungsi dari dapat ditentukan dengan cara:

$table attributes columnalign right center left columnspacing 0px end attributes row cell theta over 4 minus 2 space sin space theta times cos space theta end cell equals cell fraction numerator sin to the power of negative 1 end exponent space open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator 4 end fraction minus up diagonal strike 2 times fraction numerator 3 x over denominator up diagonal strike 2 end fraction times fraction numerator sisi space samping over denominator sisi space miring end fraction end cell row blank equals cell fraction numerator sin to the power of negative 1 end exponent space open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator 4 end fraction minus 3 x times fraction numerator square root of 4 minus 9 x squared end root over denominator 2 end fraction end cell row blank equals cell fraction numerator sin to the power of negative 1 end exponent space open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator 4 end fraction minus fraction numerator 3 x square root of 4 minus 9 x squared end root over denominator 2 end fraction end cell end table$

Jadi, $theta over 4 minus 2 space sin space theta times cos space theta equals fraction numerator sin to the power of negative 1 end exponent space open parentheses begin display style fraction numerator 3 x over denominator 2 end fraction end style close parentheses over denominator 4 end fraction minus fraction numerator 3 x square root of 4 minus 9 x squared end root over denominator 2 end fraction$ .