Untuk 0∘≤θ≤360∘, carilah nilai θ yang memenuhi masing-masing persamaan berikut. f. tan2θ∘=cosθ∘1+1

Pertanyaan

Untuk $0^\circ\leq\theta\leq360^\circ$ , carilah nilai $\theta$ yang memenuhi masing-masing persamaan berikut.

f. $\tan^2\;\theta^\circ=\frac1{\cos\;\theta^\circ}+1$

Pembahasan Soal:

Ingat bahwa:

$fraction numerator 1 over denominator cos space x end fraction equals s e c space x$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan squared space theta degree end cell equals cell fraction numerator 1 over denominator cos space theta degree end fraction plus 1 end cell row cell tan squared open parentheses theta close parentheses end cell equals cell sec open parentheses theta close parentheses plus 1 end cell row cell tan squared open parentheses theta close parentheses minus sec open parentheses theta close parentheses minus 1 end cell equals 0 row cell negative 1 minus sec open parentheses theta close parentheses plus sec squared open parentheses theta close parentheses minus 1 end cell equals 0 row cell negative 2 minus sec open parentheses theta close parentheses plus sec squared open parentheses theta close parentheses end cell equals 0 row cell sec open parentheses theta close parentheses end cell equals cell 2 comma thin space sec open parentheses theta close parentheses equals negative 1 end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell sec open parentheses theta close parentheses equals 2 end cell row blank rightwards arrow cell theta equals 60 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 60 degree end cell row blank blank blank row blank rightwards arrow cell theta equals 300 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 300 degree end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell sec open parentheses theta close parentheses equals negative 1 end cell row blank rightwards arrow cell theta equals 180 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 180 degree end cell end table$

Jadi, nilai $theta$ yang memenuhi persamaan tersebut pada interval $0 degree less or equal than theta less or equal than 360 degree$ adalah

$theta equals open curly brackets 60 degree comma thin space 300 degree comma thin space 180 degree close curly brackets$ .

Pembahasan terverifikasi oleh Roboguru

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Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 07 Oktober 2021

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Untuk 0∘≤θ≤360∘, carilah nilai θ yang memenuhi masing-masing persamaan berikut. e. 2sinθ∘cosθ∘=sinθ∘

Pembahasan Soal:

Ingat bahwa:

$sin space left parenthesis 180 minus x right parenthesis degree equals sin space x degree$
$cos space left parenthesis negative x right parenthesis degree equals cos space x degree$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell 2 sin open parentheses theta close parentheses cos open parentheses theta close parentheses equals sin open parentheses theta close parentheses 2 sin open parentheses theta close parentheses cos open parentheses theta close parentheses minus sin open parentheses theta close parentheses equals 0 sin open parentheses theta close parentheses open parentheses 2 cos open parentheses theta close parentheses minus 1 close parentheses equals 0 end cell row cell sin open parentheses theta close parentheses equals 0 space atau thin space 2 cos open parentheses theta close parentheses minus 1 equals 0 end cell end table end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell sin open parentheses theta close parentheses equals 0 end cell row blank rightwards arrow cell theta equals 0 plus 360 degree n end cell row theta equals cell 360 degree n end cell row cell Untuk space n end cell equals cell left curly bracket 0 comma space 1 right curly bracket end cell row blank blank blank row blank rightwards arrow cell theta equals 180 degree plus 360 degree n end cell row cell Untuk space n end cell equals cell left curly bracket 0 right curly bracket end cell row theta equals cell 180 degree end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell 2 cos open parentheses theta close parentheses minus 1 equals 0 end cell row cell 2 cos open parentheses theta close parentheses end cell equals 1 row cell cos open parentheses theta close parentheses end cell equals cell 1 half end cell row blank rightwards arrow cell theta equals 60 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 60 degree space end cell row blank blank blank row blank rightwards arrow cell theta equals 300 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 300 degree end cell end table$

Jadi, nilai $theta$ yang memenuhi persamaan tersebut pada interval $0 degree less or equal than theta less or equal than 360 degree$ adalah

$theta equals left curly bracket 0 degree comma thin space 180 degree comma thin space 360 degree comma thin space 60 degree comma thin space 300 degree right curly bracket$ .

Roboguru

Selesaikanlah persamaan berikut untuk 0∘<x<360∘ 3sin2x+4sinx⋅cosx−cos2x=2

Pembahasan Soal:

Dengan menggunakan rumus trigonommetri, diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 3 sin squared space x plus 4 sin space x times cos space x minus cos squared space x end cell equals 2 row cell 3 sin squared space x plus 2 times 2 sin space x times cos space x minus open parentheses 1 minus sin squared space x close parentheses end cell equals 2 row cell 3 sin squared space x plus 2 sin space 2 x minus 1 plus sin squared space x end cell equals 2 row cell open parentheses negative 2 close parentheses minus 2 sin squared space x plus 2 sin space 2 x end cell equals 3 row cell negative 2 open parentheses cos space 2 x minus 1 close parentheses plus 2 sin space 2 x end cell equals 3 row cell negative 2 cos space 2 x plus 2 plus 2 sin space 2 x end cell equals 3 row cell negative 2 cos space 2 x plus 2 sin space 2 x end cell equals 1 end table$

Kemudian misalkan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell negative 2 cos space 2 x plus 2 sin space 2 x end cell equals cell k cos left parenthesis 2 x plus alpha right parenthesis degree end cell row blank equals cell k cos space 2 x degree times cos space alpha degree minus k sin space 2 x degree times sin space alpha degree end cell end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space alpha degree end cell equals cell negative 2 end cell row cell k sin space alpha degree end cell equals cell negative 2 end cell row k equals cell square root of open parentheses negative 2 close parentheses squared plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 4 plus 4 end root end cell row blank equals cell 2 square root of 2 end cell row cell tan space alpha degree end cell equals cell fraction numerator negative 2 over denominator negative 2 end fraction space left parenthesis kuadran space III right parenthesis end cell row cell alpha degree end cell equals cell 225 degree end cell end table$

Kemudian,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space left parenthesis 2 x plus alpha right parenthesis degree end cell equals cell 2 square root of 2 cos space left parenthesis 2 x plus 225 right parenthesis degree equals 1 end cell row blank left right double arrow cell cos space left parenthesis 2 x plus 225 right parenthesis degree equals fraction numerator 1 over denominator 2 square root of 2 end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis 2 x plus 225 right parenthesis degree end cell equals cell fraction numerator 1 over denominator 2 square root of 2 end fraction end cell row cell cos space left parenthesis 2 x plus 225 right parenthesis degree end cell equals cell cos space 69 comma 3 degree end cell row cell space 2 x plus 225 degree end cell equals cell plus-or-minus 69 comma 3 degree plus n times 360 degree end cell row cell 2 x end cell equals cell negative 155 comma 7 degree plus n times 360 degree end cell row x equals cell negative 77 comma 85 degree plus n times 180 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 comma space 2 end cell row x equals cell negative 77 comma 85 degree plus 0 times 180 degree end cell row blank equals cell negative 77 comma 85 degree end cell row x equals cell negative 77 comma 85 degree plus 1 times 180 degree end cell row blank equals cell 102 comma 15 degree end cell row x equals cell negative 77 comma 85 degree plus 2 times 180 degree end cell row blank equals cell 282 comma 15 degree end cell row cell atau space 2 x end cell equals cell negative 294 comma 3 degree plus n times 360 degree end cell row x equals cell negative 147 comma 15 degree plus n times 180 degree end cell row n equals cell 0 comma space 1 comma space 2 end cell row x equals cell negative 147 comma 15 degree plus 0 times 180 degree end cell row blank equals cell negative 147 comma 15 degree end cell row x equals cell negative 147 comma 15 degree plus 1 times 180 degree end cell row blank equals cell 32 comma 85 degree end cell row x equals cell negative 147 comma 15 degree plus 2 times 180 degree end cell row blank equals cell 212 comma 85 degree end cell end table$

Dengan demikian,

$table attributes columnalign right center left columnspacing 0px end attributes row HP equals cell open curly brackets 32 comma 85 degree comma space 102 comma 15 degree comma space 212 comma 85 degree comma space 282 comma 15 degree close curly brackets end cell end table$ .

Roboguru

Selesaikanlah persamaan berikut untuk 0∘<x<360∘ cos2x−3sinx⋅cosx+2sin2x=0

Pembahasan Soal:

Dengan menggunakan rumus trigonommetri, diperoleh:

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

Kemudian misalkan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 sin space 2 x minus cos space 2 x end cell equals cell k cos left parenthesis 2 x plus alpha right parenthesis degree end cell row blank equals cell k cos space 2 x degree times cos space alpha degree minus k sin space 2 x degree times sin space alpha degree end cell end table$

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space alpha degree end cell equals cell negative 1 end cell row cell k sin space alpha degree end cell equals 3 row k equals cell square root of open parentheses negative 1 close parentheses squared plus open parentheses 3 close parentheses squared end root end cell row blank equals cell square root of 1 plus 9 end root end cell row blank equals cell square root of 10 end cell row cell tan space alpha degree end cell equals cell fraction numerator 3 over denominator negative 1 end fraction space left parenthesis kuadran space II right parenthesis end cell row cell alpha degree end cell equals cell 108 comma 43 degree end cell end table$

Kemudian,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space left parenthesis 2 x plus alpha right parenthesis degree end cell equals cell square root of 10 cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree equals negative 3 end cell row blank left right double arrow cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree equals fraction numerator negative 3 over denominator square root of 10 end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree end cell equals cell fraction numerator negative 3 over denominator square root of 10 end fraction end cell row cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree end cell equals cell cos space 161 comma 57 degree end cell row cell space 2 x plus 108 comma 43 degree end cell equals cell plus-or-minus 161 comma 57 degree plus n times 360 degree end cell row cell 2 x end cell equals cell 53 comma 14 degree plus n times 360 degree end cell row x equals cell 26 comma 57 degree plus n times 180 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 26 comma 57 degree plus 0 times 180 degree end cell row blank equals cell 26 comma 57 degree end cell row x equals cell 26 comma 57 degree plus 1 times 180 degree end cell row blank equals cell 206 comma 57 degree end cell row cell atau space 2 x end cell equals cell negative 270 degree plus n times 360 degree end cell row x equals cell negative 135 degree plus n times 180 degree end cell row n equals cell 0 comma space 1 comma space 2 end cell row x equals cell negative 135 degree plus 0 times 180 degree end cell row blank equals cell negative 135 degree end cell row x equals cell negative 135 degree plus 1 times 180 degree end cell row blank equals cell 45 degree end cell row x equals cell negative 135 degree plus 2 times 180 degree end cell row blank equals cell 225 degree end cell end table$

Dengan demikian,

$table attributes columnalign right center left columnspacing 0px end attributes row HP equals cell open curly brackets 26 comma 57 degree comma space 45 degree comma space 206 comma 57 degree comma space 225 degree close curly brackets end cell end table$ .

Roboguru

Buktikan bahwa cosx1+sinx+1+sinxcosx=cosx2 dan cari semua besar sudut antara −180∘ dan 180∘ yang merupakan solusi dari persamaan (1+sinx)2+cos2x=4cosx(1+sinx).

Pembahasan Soal:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator 1 plus sin space x end fraction end cell equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator 1 plus sin space x end fraction cross times fraction numerator 1 minus sin space x over denominator 1 minus sin space x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x space left parenthesis 1 minus sin space x right parenthesis over denominator 1 minus sin squared x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x space left parenthesis 1 minus sin space x right parenthesis over denominator cos squared x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator 1 minus sin space x over denominator cos space x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x plus 1 minus sin space x over denominator cos space x end fraction end cell row blank equals cell fraction numerator 2 over denominator cos space x end fraction space left parenthesis terbukti right parenthesis end cell end table end style$

$begin mathsize 12px style open parentheses 1 plus sin x close parentheses squared plus cos squared x equals 4 cos space x space left parenthesis 1 plus sin space x right parenthesis end style$
$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 1 plus sin x close parentheses squared plus cos squared x minus 4 space cos space x space left parenthesis 1 plus sin space x right parenthesis end cell equals 0 row cell open parentheses 1 plus sin x close parentheses squared plus cos squared x minus 4 space cos space x space left parenthesis 1 plus sin space x right parenthesis end cell equals 0 row cell 1 plus 2 space sin space x plus sin squared x plus cos squared x minus 4 space cos space x minus 4 space cos space x space sin space x end cell equals 0 row cell 1 plus 2 space sin space x plus 1 minus 4 space cos space x minus 4 space cos space x space sin space x end cell equals 0 row cell 2 space sin space x minus 4 space cos space x minus 4 space cos space x space sin space x plus 2 end cell equals 0 row cell sin space x minus 2 space cos space x minus 2 space cos space x space sin space x plus 1 end cell equals 0 row cell left parenthesis sin space x plus 1 right parenthesis left parenthesis negative 2 space cos space x plus 1 right parenthesis end cell equals 0 end table end style$

$begin mathsize 12px style sin space x equals negative 1 space atau space cos space x equals 1 half end style$

$sin space x equals negative 1$

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell negative 1 end cell row cell sin space x end cell equals cell sin space 270 degree end cell row x equals cell 270 plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 270 degree plus 0.360 end cell row blank equals cell 270 degree space left parenthesis tidak space memenuhi right parenthesis end cell row blank blank blank row x equals cell left parenthesis 180 minus 270 right parenthesis plus k.360 end cell row blank equals cell negative 90 degree plus k.360 end cell row blank blank blank row k equals 0 row x equals cell negative 90 degree plus 0.360 degree end cell row blank equals cell negative 90 degree end cell row blank blank blank row k equals 1 row x equals cell negative 90 degree plus 1.360 degree end cell row blank equals cell 270 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table$

$cos space x equals 1 half$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals cell 1 half end cell row cell cos space x end cell equals cell cos space 60 degree end cell row x equals cell 60 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 60 degree plus 0.360 degree end cell row blank equals cell 60 degree end cell row blank blank blank row x equals cell negative 60 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals negative 60 degree plus 0.360 degree end cell row blank equals cell negative 60 degree end cell end table$

Jadi, himpunan penyelesaian dari persamaan trigonometri tersebut adalah $open curly brackets negative 60 degree comma space minus 90 degree comma space 60 degree close curly brackets$ .

Roboguru

Selesaikan persamaan trigonometri di bawah ini, untuk 0∘≤x≤360∘. f. 6sinxcosx+2sinx=1+3cosx

Pembahasan Soal:

Ingat!

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell sin space alpha end cell row x equals cell alpha plus k.360 degree end cell row blank blank atau row x equals cell left parenthesis 180 minus alpha right parenthesis plus k.360 degree end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals cell cos space alpha end cell row x equals cell alpha plus k.360 degree end cell row blank blank atau row x equals cell negative alpha plus k.360 degree end cell end table$

Maka:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 6 space sin space x space cos space x plus 2 space sin space x end cell equals cell 1 plus 3 space cos x end cell row cell 6 space sin space x space cos space x plus 2 space sin space x minus 3 space cos space x minus 1 end cell equals 0 row cell left parenthesis 3 space cos space x plus 1 right parenthesis left parenthesis 2 space sin space x minus 1 right parenthesis end cell equals 0 end table end style$

$begin mathsize 12px style cos space x equals negative 1 third space atau space sin space x equals 1 half end style$

$cos space x equals negative 1 third$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals cell negative 1 third end cell row cell cos space x end cell equals cell cos space 109 comma 47 degree end cell row x equals cell 109 comma 47 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 109 comma 47 degree plus 0.360 degree end cell row blank equals cell 109 comma 47 degree end cell row blank blank blank row x equals cell negative 109 comma 47 degree plus k.360 degree end cell row blank blank blank row k equals 1 row blank rightwards double arrow cell x equals negative 109 comma 47 degree plus 1.360 degree end cell row blank equals cell 250 comma 53 degree end cell end table$

$sin space x equals 1 half$

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell 1 half end cell row cell sin space x end cell equals cell sin space 30 degree end cell row x equals cell 30 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 30 degree plus 0.360 degree end cell row x equals cell 30 degree end cell row blank blank blank row x equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus k.360 degree end cell row blank equals cell 150 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 150 degree plus 0.360 degree end cell row blank equals cell 150 degree end cell end table$

Jadi, himpunan penyelesaian dari persamaan trigonometri tersebut adalah $open curly brackets 30 degree space semicolon space 109 comma 47 degree space semicolon space 150 degree space semicolon space 250 comma 53 degree close curly brackets$ .

Roboguru