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7. Jika m tan (θ−30∘)=n tan (θ+12∘), tunjukkan bahwa cos 2θ=2(m−n)m+n​.

Pertanyaan

7. Jika m space tan space open parentheses theta minus 30 degree close parentheses equals n space tan space open parentheses theta plus 12 degree close parentheses, tunjukkan bahwa cos space 2 theta equals fraction numerator m plus n over denominator 2 open parentheses m minus n close parentheses end fraction.space 

I. Ridha

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

telah ditunjukkan bahwa cos space 2 theta equals fraction numerator m plus n over denominator 2 open parentheses m minus n close parentheses end fraction.space 

Pembahasan

Perhatian!
Persamaan pada soal diralat menjadi: m space tan space open parentheses theta minus 30 degree close parentheses equals n space tan space open parentheses theta plus 120 degree close parentheses

Ingat!

  • tan space open parentheses 90 degree plus alpha close parentheses equals negative text cot end text space alpha 
  • text cot end text space alpha equals fraction numerator 1 over denominator tan space alpha end fraction
  • cos space open parentheses alpha plus beta close parentheses equals cos space alpha times cos space beta minus sin space alpha times sin space beta 
  • cos space open parentheses alpha minus beta close parentheses equals cos space alpha times cos space beta plus sin space alpha times sin space beta 
  • 1 equals a over a, dengan a not equal to 0
  • ​​​​​cos space open parentheses negative 60 degree close parentheses equals 1 half

Oleh karena itu, dapat diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell m space tan space open parentheses theta minus 30 degree close parentheses end cell equals cell n space tan space open parentheses theta plus 120 degree close parentheses end cell row cell m space tan space open parentheses theta minus 30 degree close parentheses end cell equals cell n space tan space open parentheses 90 degree plus open parentheses theta plus 30 degree close parentheses close parentheses end cell row cell m space tan space open parentheses theta minus 30 degree close parentheses end cell equals cell n open parentheses negative text cot end text open parentheses theta plus 30 degree close parentheses close parentheses end cell row cell m space tan space open parentheses theta minus 30 degree close parentheses end cell equals cell n open parentheses fraction numerator negative 1 over denominator tan space open parentheses theta plus 30 degree close parentheses end fraction close parentheses end cell row cell m space tan space open parentheses theta minus 30 degree close parentheses end cell equals cell fraction numerator negative n over denominator tan space open parentheses theta plus 30 degree close parentheses end fraction end cell row cell tan space open parentheses theta minus 30 degree close parentheses times tan space open parentheses theta plus 30 degree close parentheses end cell equals cell fraction numerator negative n over denominator m end fraction end cell row cell fraction numerator sin space open parentheses theta minus 30 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses end fraction times fraction numerator sin space open parentheses theta plus 30 degree close parentheses over denominator cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator negative n over denominator m end fraction end cell row cell fraction numerator sin space open parentheses theta minus 30 degree close parentheses times sin space open parentheses theta plus 30 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator negative n over denominator m end fraction end cell end table

Misalkan A equals theta minus 30 degree dan B equals theta plus 30 degree, maka dapat diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative n over denominator m end fraction space space space space space left parenthesis 1 right parenthesis end cell end table

dan saat kedua ruas pada (1) dikurang dengan 1, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space open parentheses A close parentheses times sin space open parentheses B close parentheses over denominator cos space open parentheses A close parentheses times cos space open parentheses B close parentheses end fraction minus 1 end cell equals cell fraction numerator negative n over denominator m end fraction minus 1 end cell row cell fraction numerator sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction minus fraction numerator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative n over denominator m end fraction minus m over m end cell row cell fraction numerator sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis minus cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative n minus m over denominator m end fraction end cell row cell fraction numerator negative cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis plus sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative m minus n over denominator m end fraction end cell row cell fraction numerator negative open parentheses cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis minus sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis close parentheses over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative open parentheses m plus n close parentheses over denominator m end fraction end cell row cell fraction numerator cos space open parentheses A plus B close parentheses over denominator cos space open parentheses A close parentheses times cos space open parentheses B close parentheses end fraction end cell equals cell fraction numerator m plus n over denominator m end fraction end cell row blank blank cell blank presuperscript table attributes columnalign right end attributes row cell text karena end text space A equals theta minus 30 degree end cell row cell B equals theta plus 30 degree end cell end table end presuperscript end cell row blank blank cell blank presuperscript text  maka end text end presuperscript end cell row cell fraction numerator cos space open parentheses theta minus 30 degree plus theta plus 30 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator m plus n over denominator m end fraction end cell row cell fraction numerator cos space open parentheses 2 theta close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator m plus n over denominator m end fraction space space space space space space space space space space space space space left parenthesis 2 right parenthesis end cell end table  

dan saat kedua ruas pada (1) ditambah dengan 1, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space open parentheses A close parentheses times sin space open parentheses B close parentheses over denominator cos space open parentheses A close parentheses times cos space open parentheses B close parentheses end fraction plus 1 end cell equals cell fraction numerator negative n over denominator m end fraction plus 1 end cell row cell fraction numerator sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction plus fraction numerator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative n over denominator m end fraction plus m over m end cell row cell fraction numerator sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis plus cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator negative n plus m over denominator m end fraction end cell row cell fraction numerator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis plus sin space left parenthesis A right parenthesis times sin space left parenthesis B right parenthesis over denominator cos space left parenthesis A right parenthesis times cos space left parenthesis B right parenthesis end fraction end cell equals cell fraction numerator m minus n over denominator m end fraction end cell row cell fraction numerator cos space open parentheses A minus B close parentheses over denominator cos space open parentheses A close parentheses times cos space open parentheses B close parentheses end fraction end cell equals cell fraction numerator m minus n over denominator m end fraction end cell row blank blank cell blank presuperscript table attributes columnalign right end attributes row cell text karena end text space A equals theta minus 30 degree end cell row cell B equals theta plus 30 degree end cell end table end presuperscript end cell row blank blank cell blank presuperscript text  maka end text end presuperscript end cell row cell fraction numerator cos space open parentheses theta minus 30 degree minus open parentheses theta plus 30 degree close parentheses close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator m minus n over denominator m end fraction end cell row cell fraction numerator cos space open parentheses theta minus 30 degree minus theta minus 30 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator m minus n over denominator m end fraction end cell row cell fraction numerator cos space open parentheses negative 60 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end cell equals cell fraction numerator m minus n over denominator m end fraction space space space space space space space space space space space space space left parenthesis 3 right parenthesis end cell end table

Dengan membagi (2) dengan (3), diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator fraction numerator cos space 2 theta over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction over denominator fraction numerator cos space open parentheses negative 60 degree close parentheses over denominator cos space open parentheses theta minus 30 degree close parentheses times cos space open parentheses theta plus 30 degree close parentheses end fraction end fraction end cell equals cell fraction numerator fraction numerator m plus n over denominator m end fraction over denominator fraction numerator m minus n over denominator m end fraction end fraction end cell row cell fraction numerator cos space 2 theta over denominator cos space open parentheses negative 60 degree close parentheses end fraction end cell equals cell fraction numerator m plus n over denominator m minus n end fraction end cell row cell cos space 2 theta end cell equals cell cos space open parentheses negative 60 degree close parentheses times fraction numerator m plus n over denominator m minus n end fraction end cell end table

dan karena cos space open parentheses negative 60 degree close parentheses equals 1 half, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 theta end cell equals cell 1 half times fraction numerator m plus n over denominator m minus n end fraction end cell row cell cos space 2 theta end cell equals cell fraction numerator m plus n over denominator 2 open parentheses m minus n close parentheses end fraction end cell end table  

Dengan demikian, telah ditunjukkan bahwa cos space 2 theta equals fraction numerator m plus n over denominator 2 open parentheses m minus n close parentheses end fraction.space 

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