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Tunjukan bahwa: f.  cotan2θ​=cosecθ+cotanθ

Pertanyaan

Tunjukan bahwa:

f.  cotan2θ=cosecθ+cotanθ   

Pembahasan Soal:

Ingat kembali rumus:

tan2A=1+cosAsinA  

cosecA=sinA1cotanA=tanA1

Akan ditunjukkan bahwa, cotan2θ=cosecθ+cotanθ.

  cotan2θ=====tan2θ11+cosθsinθ1sinθ1+cosθsinθ1+sinθcosθcosecθ+cotanθ(terbukti)

Jadi, terbukti bahwa, cotan2θ=cosecθ+cotanθ.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 31 Agustus 2021

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Pertanyaan yang serupa

Bentuk sederhana dari  adalah

Pembahasan Soal:

Ingat kembali bahwa:

table attributes columnalign right center left columnspacing 0px end attributes row cell c o t space alpha end cell equals cell fraction numerator 1 over denominator tan space alpha end fraction end cell row cell tan space 1 half alpha end cell equals cell fraction numerator sin space alpha over denominator 1 plus cos space alpha end fraction end cell end table

Misalkan: cos space 78 degree minus sin space 78 degree times text cot end text space 39 degree equals space A dengan menggunakan rumus di atas, maka bentuk sederha dari A adalah

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell cos space 78 degree minus sin space 78 degree times text cot end text space open parentheses fraction numerator 78 degree over denominator 2 end fraction close parentheses end cell row blank equals cell cos space 78 degree minus sin space 78 degree times fraction numerator 1 over denominator tan space open parentheses fraction numerator 78 degree over denominator 2 end fraction close parentheses end fraction end cell row blank equals cell cos space 78 degree minus sin space 78 degree times fraction numerator 1 over denominator begin display style fraction numerator sin space 78 degree over denominator 1 plus cos space 78 degree end fraction end style end fraction end cell row blank equals cell cos space 78 degree minus sin space 78 degree times fraction numerator 1 plus cos space 78 degree over denominator sin space 78 degree end fraction end cell row blank equals cell cos space 78 degree minus open parentheses 1 plus cos space 78 degree close parentheses end cell row blank equals cell cos space 78 degree minus 1 minus cos space 78 degree end cell row A equals cell negative 1 end cell end table

Jadi, bentuk sederhana dari cos space 78 degree minus sin space 78 degree times text cot end text space 39 degree adalah negative 1. 

Oleh karena itu, jawaban yang benar adalah D.

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Roboguru

Tunjukan bahwa: tan2A​=cosecA−cotanA

Pembahasan Soal:

Ingat kembali rumus:

tan2A=sinA1cosA 

cosecA=sinA1cotanA=sinAcosA

Akan ditunjukkan bahwa, tan2A=cosecAcotanA

tan2Atan2Atan2A===sinA1cosAsinA1sinAcosAcosecAcotanAterbukti 

Jadi, terbukti bahwa, tan2A=cosecAcotanA.

0

Roboguru

Tentukan nilai x yang memenuhi persamaan berikut untuk 0∘≤x≤360∘. b. (cot(2x−60)∘+3​)(sec3x−2​)=0

Pembahasan Soal:

Ingat kembali:

  • cotx=tanx1 
  • secx=cosx1 
  • cosx=cosα{x=α+k360x=α+k360
  • tanx=tanαx=α+k180 

Diberikan persamaan (cot(2x60)+3)(sec3x2)=0 untuk 0x360. Maka:

- Untuk (cot(2x60)+3)=0 

(cot(2x60)+3)=0cot(2x60)+3=0cot(2x60)=3tan(2x60)1=331=tan(2x60)tan(2x60)=31tan(2x60)=tan150 

  • Persamaan tangen

2x602xx===150+k180210+k180105+k90 

k equals 0 rightwards arrow table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 90 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table 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 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 90 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row cell degree plus 90 degree end cell equals cell 195 degree end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table 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 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 90 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 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 105 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 180 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 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 285 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table  

- Untuk (sec3x2)=0 

(sec3x2)sec3x2sec3xcos3x121cos3xcos3xcos3x========0022cos3x21212cos45 

  • Persamaan cosinus (1)

3xx==45+k36015+k120 

k=0x=15+0120=15+0=15k=1x=15+1120=15+120=135k=2x=15+2120=15+240=255 

  • Persamaan cosinus (2)

3xx==45+k36015+k120 

k=1x=15+1120=15+120=105k=2x=15+2120=15+240=225k=3x=15+3120=15+360=345 

Jadi, nilai x yang memenuhi persamaan (cot(2x60)+3)(sec3x2)=0 untuk 0x360 adalah {15,105,135,195,225,255,285,345}.

0

Roboguru

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space x end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell s e c space x end cell equals cell fraction numerator 1 over denominator cos space x end fraction end cell row cell sin squared x plus cos squared x end cell equals 1 end table

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of fraction numerator s e c space x minus tan space x over denominator s e c space x plus tan space x end fraction end root end cell equals cell square root of fraction numerator begin display style fraction numerator 1 over denominator cos space x end fraction end style minus begin display style fraction numerator sin space x over denominator cos space x end fraction end style over denominator begin display style fraction numerator 1 over denominator cos space x end fraction end style plus begin display style fraction numerator sin space x over denominator cos space x end fraction end style end fraction end root end cell row blank equals cell square root of fraction numerator begin display style fraction numerator 1 minus sin space x over denominator cos space x end fraction end style over denominator begin display style fraction numerator 1 plus sin space x over denominator cos space x end fraction end style end fraction end root end cell row blank equals cell square root of fraction numerator 1 minus sin space x over denominator 1 plus sin space x end fraction end root cross times square root of fraction numerator 1 minus sin space x over denominator 1 minus sin space x end fraction end root end cell row blank equals cell fraction numerator 1 minus sin space x over denominator square root of 1 minus sin squared x end root end fraction end cell row blank equals cell fraction numerator 1 minus sin space x over denominator square root of 1 minus left parenthesis 1 minus cos squared x right parenthesis end root end fraction end cell row blank equals cell fraction numerator 1 minus sin space x over denominator square root of cos squared x end root end fraction end cell row blank equals cell fraction numerator 1 minus sin space x over denominator cos space x end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space x end fraction minus fraction numerator sin space x over denominator cos space x end fraction end cell row blank equals cell s e c space x minus tan space x end cell end table

Oleh karena itu, jawaban yang benar adalah D.

1

Roboguru

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space x end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell c o t space x end cell equals cell fraction numerator cos space x over denominator sin space x end fraction end cell row cell c s c space x end cell equals cell fraction numerator 1 over denominator sin space x end fraction end cell row cell s e c space x end cell equals cell fraction numerator 1 over denominator cos space x end fraction end cell row cell sin squared x plus cos squared x end cell equals 1 end table

Sehingga,

left parenthesis 1 plus c o t space x minus c s c space x right parenthesis left parenthesis 1 plus tan space x plus s e c space x right parenthesis equals open parentheses 1 plus fraction numerator cos space x over denominator sin space x end fraction minus fraction numerator 1 over denominator sin space x end fraction close parentheses open parentheses 1 plus fraction numerator sin space x over denominator cos space x end fraction plus fraction numerator 1 over denominator cos space x end fraction close parentheses equals open parentheses fraction numerator sin space x plus cos space x minus 1 over denominator sin space x end fraction close parentheses open parentheses fraction numerator cos space x plus sin space x plus 1 over denominator cos space x end fraction close parentheses equals fraction numerator open parentheses sin space x plus cos space x close parentheses squared minus 1 over denominator sin space x times cos space x end fraction equals fraction numerator sin squared x plus cos squared x plus 2 sin space x times cos space x minus 1 over denominator sin space x times cos space x end fraction equals fraction numerator 1 plus 2 sin space x times cos space x minus 1 over denominator sin space x times cos space x end fraction equals fraction numerator 2 space sin space x times cos space x over denominator sin space x times cos space x end fraction equals 2

Oleh karena itu, jawaban yang benar adalah C.

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