Himpunan penyelesaian dari persamaan sin(x+30)∘+sin(x−120)∘=0 untuk 0∘≤x≤360∘ adalah...

Pertanyaan

Himpunan penyelesaian dari persamaan $\sin\;\left(x+30\right)^\circ+\sin\;\left(x-120\right)^\circ=0$ untuk $0^\circ\leq x\leq360^\circ$ adalah...

$open curly brackets 45 degree comma space 135 degree close curly brackets$
$open curly brackets 45 degree comma space 225 degree close curly brackets$
$open curly brackets 45 degree comma space 315 degree close curly brackets$
$open curly brackets 135 degree comma space 225 degree close curly brackets$
$open curly brackets 135 degree comma space 315 degree close curly brackets$

Pembahasan Soal:

Ingat!

$\sin\;\alpha+\sin\;\beta=2\;\sin\;\left(\frac{\alpha+\beta}2\right)\cdot\cos\;\left(\frac{\alpha-\beta}2\right)$

dan

$\begin{array}{rcl}\sin\;x&=&\sin\;\alpha\\x_1&=&\alpha+k\cdot360^\circ\\x_2&=&\left(180-\alpha\right)+k\cdot360^\circ\end{array}$

Dengan menggunakan rumus trigonometri, diperoleh:

$\style{font-size:12px}{\begin{array}{rcl}\sin\;\left(x+30\right)^\circ+\sin\;\left(x-120\right)^\circ&=&0\\2\sin\left(\frac{x+30^\circ+x-120^\circ}2\right)\cdot\cos\left(\frac{x+30^\circ-x+120^\circ}2\right)&=&0\\2\sin\left(x-45\right)^\circ\cdot\cos\;75^\circ&=&0\\\sin\left(x-45\right)^\circ&=&0\\\sin\left(x-45\right)^\circ&=&\sin\;0^\circ\end{array}}$

Maka

$\begin{array}{rcl}x_1-45^\circ&=&0^\circ+k\cdot360\\x_1&=&45^\circ+k\cdot360\\x_2-45&=&\left(180^\circ-0^\circ\right)+k\cdot360^\circ\\x_2&=&180^\circ+45^\circ+k\cdot360^\circ\\x_2&=&225^\circ+k\cdot360\end{array}$

Untuk $k=0$ , maka:

$\begin{array}{rcl}x_1&=&45^\circ+0\cdot360^\circ=45^\circ\\x_2&=&225^\circ+0\cdot360^\circ=225^\circ\end{array}$

Untuk $k=1$ , maka:

$\begin{array}{rcl}x_1&=&45^\circ+1\cdot360^\circ=405^\circ\\x_2&=&225^\circ+1\cdot360^\circ=585^\circ\\&&\text{Tidak Memenuhi}\end{array}$

Dengan demikian,

$HP equals open curly brackets 45 degree comma space 225 degree close curly brackets$

Oleh karena itu, jawaban yang benar adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Wati

Terakhir diupdate 07 Oktober 2021

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Dengan menggunakan rumus trigonometri, diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses 3 x minus 45 close parentheses degree plus sin space open parentheses x minus 15 close parentheses degree end cell equals 0 row cell 2 sin open parentheses fraction numerator 3 x minus 45 plus x minus 15 over denominator 2 end fraction close parentheses times sin open parentheses fraction numerator 3 x minus 45 minus x plus 15 over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 sin open parentheses 2 x minus 30 close parentheses times sin open parentheses x minus 15 close parentheses end cell equals 0 row cell sin open parentheses 2 x minus 30 close parentheses times sin open parentheses x minus 15 close parentheses end cell equals 0 end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis 2 x minus 30 right parenthesis end cell equals 0 row cell sin space left parenthesis 2 x minus 30 right parenthesis end cell equals cell sin space 0 degree end cell row cell 2 x minus 30 end cell equals cell 0 degree plus n times 360 degree end cell row cell 2 x end cell equals cell 30 degree plus n times 360 degree end cell row x equals cell 15 degree plus n times 180 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 15 degree plus 0 times 180 degree end cell row blank equals cell 15 degree end cell row x equals cell 15 degree plus 1 times 180 degree end cell row blank equals cell 195 degree end cell row cell atau space 2 x minus 30 end cell equals cell open parentheses 180 minus 0 close parentheses degree plus n times 360 degree end cell row blank equals cell 180 degree plus n times 360 degree end cell row cell 2 x end cell equals cell 210 degree plus n times 360 degree end cell row x equals cell 105 degree plus n times 180 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 105 degree plus 0 times 180 degree end cell row blank equals cell 105 degree end cell row x equals cell 105 degree plus 1 times 180 degree end cell row blank equals cell 285 degree end cell end table$

atau

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis x minus 15 right parenthesis end cell equals 0 row cell sin space left parenthesis x minus 15 right parenthesis end cell equals cell sin space 0 degree end cell row cell x minus 15 end cell equals cell 0 degree plus n times 360 degree end cell row x equals cell 15 degree plus n times 360 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 15 degree plus 0 times 360 degree end cell row blank equals cell 15 degree end cell row x equals cell 15 degree plus 1 times 360 degree end cell row blank equals cell 375 degree end cell row cell atau space x minus 15 end cell equals cell open parentheses 180 minus 0 close parentheses degree plus n times 360 degree end cell row blank equals cell 195 degree plus n times 360 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 195 degree plus 0 times 360 degree end cell row blank equals cell 195 degree end cell row x equals cell 195 degree plus 1 times 360 degree end cell row blank equals cell 555 degree end cell end table$

Dengan demikian,

$HP equals open curly brackets 15 degree comma space 105 degree comma space 195 degree comma space 285 degree close curly brackets$

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Pembahasan Soal:

Dengan menggunakan rumus trigonometri, diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses 3 x minus 15 close parentheses degree plus sin open parentheses 3 x minus 45 close parentheses degree end cell equals 0 row cell 2 sin open parentheses fraction numerator 3 x minus 15 plus 3 x minus 45 over denominator 2 end fraction close parentheses sin open parentheses fraction numerator 3 x minus 15 minus 3 x plus 45 over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 sin open parentheses 3 x minus 30 close parentheses sin open parentheses 15 close parentheses end cell equals 0 row cell sin open parentheses 3 x minus 30 close parentheses end cell equals 0 end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin left parenthesis 3 x minus 30 right parenthesis end cell equals 0 row cell sin left parenthesis 3 x minus 30 right parenthesis end cell equals cell sin space 0 degree end cell row cell 3 x minus 30 degree end cell equals cell space 0 degree plus n times 360 degree end cell row cell 3 x end cell equals cell 30 degree plus n times 360 degree end cell row x equals cell 10 degree plus n times 120 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 comma space 2 end cell row x equals cell 10 degree plus 0 times 120 degree end cell row blank equals cell 10 degree end cell row x equals cell 10 degree plus 1 times 120 degree end cell row blank equals cell 130 degree end cell row x equals cell 10 degree plus 2 times 120 degree end cell row blank equals cell 250 degree end cell row cell atau space 3 x minus 30 degree end cell equals cell space 180 degree plus n times 360 degree end cell row cell 3 x end cell equals cell 210 degree plus n times 360 degree end cell row x equals cell 70 degree plus n times 120 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 comma space 2 end cell row x equals cell 70 degree plus 0 times 120 degree end cell row blank equals cell 70 degree end cell row x equals cell 70 degree plus 1 times 120 degree end cell row blank equals cell 190 degree end cell row x equals cell 70 degree plus 2 times 120 degree end cell row blank equals cell 310 degree end cell end table$

Dengan demikian,

$table attributes columnalign right center left columnspacing 0px end attributes row HP equals cell open curly brackets 10 degree comma space 70 degree comma space 130 degree comma space 190 degree comma space 250 degree comma space 310 degree close curly brackets end cell row cell Hasil space penjumlahan end cell equals cell 10 degree plus 70 degree plus 130 degree plus 190 degree plus 250 degree plus 310 degree end cell row blank equals cell 960 degree end cell end table$

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Dengan menggunakan rumus trigonometri, diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses x plus 120 close parentheses degree minus sin space open parentheses x minus 60 close parentheses degree end cell equals 0 row cell 2 cos open parentheses fraction numerator x plus 120 degree plus x minus 60 degree over denominator 2 end fraction close parentheses times sin open parentheses fraction numerator x plus 120 degree minus x plus 60 degree over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 cos left parenthesis x plus 30 degree right parenthesis times sin space 90 degree end cell equals 0 row cell 2 cos left parenthesis x plus 30 degree right parenthesis times 1 end cell equals 0 row cell cos left parenthesis x plus 30 degree right parenthesis end cell equals 0 row cell cos left parenthesis x plus 30 degree right parenthesis end cell equals cell cos space 90 degree end cell end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 30 degree end cell equals cell 90 degree plus n times 360 degree end cell row x equals cell 60 degree plus n times 360 degree end cell row n equals cell 0 comma space 1 end cell row x equals cell 60 degree plus 0 times 360 degree end cell row blank equals cell 60 degree end cell row x equals cell 60 degree plus 1 times 360 degree end cell row blank equals cell 420 degree end cell end table$

atau

$table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 30 degree end cell equals cell negative 90 degree plus n times 360 degree end cell row x equals cell negative 120 degree plus n times 360 degree end cell row n equals cell 0 comma space 1 end cell row x equals cell negative 120 degree plus 0 times 360 degree end cell row blank equals cell negative 120 degree end cell row x equals cell negative 120 degree plus 1 times 360 degree end cell row blank equals cell 240 degree end cell end table$

Dengan demikian,

$HP equals open curly brackets 60 degree comma space 240 degree close curly brackets$

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$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x degree plus sin space 5 x degree end cell equals cell sin space 3 x degree end cell row cell sin space x degree plus sin space 5 x degree minus sin space 3 x degree end cell equals 0 row cell 2 sin open parentheses fraction numerator x degree plus 5 x degree over denominator 2 end fraction close parentheses cos open parentheses fraction numerator x degree minus 5 x degree over denominator 2 end fraction close parentheses minus sin space 3 x degree end cell equals 0 row cell 2 sin space 3 x degree times cos left parenthesis negative 2 x degree right parenthesis minus sin space 3 x degree end cell equals 0 row cell sin space 3 x degree open parentheses 2 cos left parenthesis negative 2 x degree right parenthesis minus 1 close parentheses end cell equals 0 row cell sin space 3 x degree end cell equals cell 0 space atau end cell row cell 2 cos left parenthesis negative 2 x degree right parenthesis minus 1 end cell equals 0 row cell cos left parenthesis negative 2 x degree right parenthesis end cell equals cell 1 half end cell end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 3 x degree end cell equals 0 row cell sin space 3 x degree end cell equals cell sin space 0 degree end cell row cell 3 x degree end cell equals cell 0 degree plus n times 360 degree end cell row cell x degree end cell equals cell 0 degree plus n times 120 degree end cell row n equals cell 0 comma space 1 comma space 2 comma space 3 end cell row cell x degree end cell equals cell 0 degree plus 0 times 120 degree end cell row blank equals cell 0 degree end cell row cell x degree end cell equals cell 0 degree plus 1 times 120 degree end cell row blank equals cell 360 degree end cell row cell x degree end cell equals cell 0 degree plus 2 times 120 degree end cell row blank equals cell 240 degree end cell row cell x degree end cell equals cell 0 degree plus 3 times 120 degree end cell row blank equals cell 360 degree end cell row blank blank atau row cell 3 straight x degree end cell equals cell 180 degree plus straight n times 360 degree end cell row cell straight x degree end cell equals cell 60 degree plus straight n times 120 degree end cell row straight n equals cell 0 comma space 1 comma space 2 end cell row cell straight x degree end cell equals cell 60 degree plus 0 times 120 degree end cell row blank equals cell 60 degree end cell row cell straight x degree end cell equals cell 60 degree plus 1 times 120 degree end cell row blank equals cell 180 degree end cell row cell straight x degree end cell equals cell 60 degree plus 2 times 120 degree end cell row blank equals cell 300 degree end cell end table$

atau

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis negative 2 x degree right parenthesis end cell equals cell cos space 2 x degree end cell row cell cos space 2 x degree end cell equals cell 1 half end cell row cell cos space 2 x degree end cell equals cell cos space 60 degree end cell row cell 2 x degree end cell equals cell 60 degree plus n times 360 degree end cell row cell x degree end cell equals cell 30 degree plus n times 180 degree end cell row n equals cell 0 comma space 1 end cell row cell x degree end cell equals cell 30 degree plus 0 times 180 degree end cell row blank equals cell 30 degree end cell row cell x degree end cell equals cell 30 degree plus 1 times 180 degree end cell row blank equals cell 210 degree end cell row blank blank atau row cell x degree end cell equals cell negative 30 degree plus n times 180 degree end cell row n equals cell 0 comma space 1 comma space 2 end cell row cell x degree end cell equals cell negative 30 degree plus 0 times 180 degree end cell row blank equals cell negative 30 degree end cell row cell x degree end cell equals cell negative 30 degree plus 1 times 180 degree end cell row blank equals cell 150 degree end cell row cell x degree end cell equals cell negative 30 degree plus 2 times 180 degree end cell row blank equals cell 330 degree end cell end table$

Dengan demikian,

$HP equals open curly brackets 0 degree comma space 30 degree comma space 60 degree comma space 120 degree comma space 150 degree comma space 180 degree comma space 210 degree comma space 240 degree comma space 300 degree comma space 330 degree comma space 360 degree close curly brackets$

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$table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses straight x plus 3 over 4 straight pi close parentheses plus sin open parentheses straight x minus 3 over 4 straight pi close parentheses end cell equals cell 1 half square root of 2 end cell row cell 2 sin space x times cos space open parentheses 3 over 4 straight pi close parentheses end cell equals cell 1 half square root of 2 end cell row cell 2 sin space x times cos space 135 degree end cell equals cell 1 half square root of 2 end cell row cell 2 sin space x times open parentheses negative 1 half square root of 2 close parentheses end cell equals cell 1 half square root of 2 end cell row cell 2 sin space x end cell equals cell fraction numerator 1 half square root of 2 over denominator negative 1 half square root of 2 end fraction end cell row cell 2 sin space x end cell equals cell negative 1 end cell row cell sin space x end cell equals cell negative 1 half end cell row cell sin space x end cell equals cell sin space 210 degree end cell end table$

maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 end cell equals cell 7 over 6 straight pi plus n times 2 straight pi end cell row blank blank atau row cell x subscript 2 end cell equals cell open parentheses straight pi minus 7 over 6 straight pi close parentheses plus n times 2 straight pi end cell row cell untuk space n end cell equals 0 row cell x subscript 1 end cell equals cell 7 over 6 straight pi plus 0 times 2 straight pi end cell row blank equals cell 7 over 6 straight pi end cell row cell x subscript 2 end cell equals cell open parentheses straight pi minus 7 over 6 straight pi close parentheses plus 0 times 2 straight pi end cell row blank equals cell negative 1 over 6 space left parenthesis tidak space memenuhi right parenthesis end cell row cell untuk space n end cell equals 1 row cell x subscript 1 end cell equals cell 7 over 6 straight pi plus 1 times 2 straight pi end cell row blank equals cell 19 over 6 straight pi space left parenthesis tidak space memenuhi right parenthesis end cell row cell x subscript 2 end cell equals cell open parentheses straight pi minus 7 over 6 straight pi close parentheses plus 1 times 2 straight pi end cell row blank equals cell 11 over 6 straight pi end cell end table$

Dengan demikian,

$HP equals open curly brackets 7 over 6 straight pi comma space 11 over 6 straight pi close curly brackets$

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