Roboguru

1. Jika tanβ=1+tanα⋅tanγtanα+tanγ​, tunjukkan bahwa: sin2β=1+sin2α⋅sin2γsin2α+sin2γ​.

Pertanyaan

1. Jika tan space beta equals fraction numerator tan space alpha plus tan space gamma over denominator 1 plus tan space alpha times tan space gamma end fraction, tunjukkan bahwa: sin space 2 beta equals fraction numerator sin space 2 alpha plus sin space 2 gamma over denominator 1 plus sin space 2 alpha times sin space 2 gamma end fraction.space 

Pembahasan Soal:

Ingat kembali:

  • tan space A equals fraction numerator sin space A over denominator cos space A end fraction
  • table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses A plus B close parentheses end cell equals cell sin space A times cos space B plus cos space A times sin space B end cell end table
  • table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses A minus B close parentheses end cell equals cell sin space A times cos space B minus cos space A times sin space B end cell end table
  • table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses A plus B close parentheses end cell equals cell cos space A times cos space B minus sin space A times sin space B end cell end table
  • table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses A minus B close parentheses end cell equals cell cos space A times cos space B plus sin space A times sin space B end cell end table 
  • sin space 2 A equals fraction numerator 2 times space tan space A over denominator 1 plus tan squared space A end fraction
  • 2 times sin space A times cos space B equals sin space open parentheses A plus B close parentheses plus sin space open parentheses A minus B close parentheses 
  • 2 times cos squared space A equals 1 plus cos space 2 A
  • 2 times sin squared space A equals 1 minus cos space 2 A

Dengan menggunakan rumus-rumus di atas, dapat diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space beta end cell equals cell fraction numerator tan space alpha plus tan space gamma over denominator 1 plus tan space alpha times tan space gamma end fraction end cell row blank equals cell fraction numerator fraction numerator sin space alpha over denominator cos space alpha end fraction plus fraction numerator sin space gamma over denominator cos space gamma end fraction over denominator 1 plus fraction numerator sin space alpha over denominator cos space alpha end fraction times fraction numerator sin space gamma over denominator cos space gamma end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator sin space alpha times cos space gamma over denominator cos space alpha times cos space gamma end fraction plus fraction numerator cos space alpha times sin space gamma over denominator cos space alpha times cos space gamma end fraction over denominator fraction numerator cos space alpha times cos space gamma over denominator cos space alpha times cos space gamma end fraction plus fraction numerator sin space alpha times sin space gamma over denominator cos space alpha times cos space gamma end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator sin space alpha times cos space gamma plus cos space alpha times sin space gamma over denominator cos space alpha times cos space gamma end fraction over denominator fraction numerator cos space alpha times cos space gamma plus sin space alpha times sin space gamma over denominator cos space alpha times cos space gamma end fraction end fraction end cell row blank equals cell fraction numerator sin space alpha times cos space gamma plus cos space alpha times sin space gamma over denominator cos space alpha times cos space gamma plus sin space alpha times sin space gamma end fraction end cell row blank equals cell fraction numerator sin space open parentheses alpha plus gamma close parentheses over denominator cos space open parentheses alpha minus gamma close parentheses end fraction end cell end table

sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 beta end cell equals cell fraction numerator 2 times tan space beta over denominator 1 plus tan squared space beta end fraction end cell row blank equals cell fraction numerator 2 times fraction numerator sin space open parentheses alpha plus gamma close parentheses over denominator cos space open parentheses alpha minus gamma close parentheses end fraction over denominator 1 plus open parentheses fraction numerator sin space open parentheses alpha plus gamma close parentheses over denominator cos space open parentheses alpha minus gamma close parentheses end fraction close parentheses squared end fraction end cell row blank equals cell fraction numerator 2 times fraction numerator sin space open parentheses alpha plus gamma close parentheses over denominator cos space open parentheses alpha minus gamma close parentheses end fraction over denominator fraction numerator cos squared space open parentheses alpha minus gamma close parentheses over denominator cos squared space open parentheses alpha minus gamma close parentheses end fraction plus fraction numerator sin squared space open parentheses alpha plus gamma close parentheses over denominator cos squared space open parentheses alpha minus gamma close parentheses end fraction end fraction end cell row blank equals cell fraction numerator 2 times fraction numerator sin space open parentheses alpha plus gamma close parentheses over denominator cos space open parentheses alpha minus gamma close parentheses end fraction over denominator fraction numerator cos squared space open parentheses alpha minus gamma close parentheses plus sin squared space open parentheses alpha plus gamma close parentheses over denominator cos squared space open parentheses alpha minus gamma close parentheses end fraction end fraction end cell row blank equals cell fraction numerator 2 times sin space open parentheses alpha plus gamma close parentheses over denominator up diagonal strike cos space open parentheses alpha minus gamma close parentheses end strike to the power of 1 end fraction cross times fraction numerator up diagonal strike cos squared space open parentheses alpha minus gamma close parentheses end strike to the power of cos space open parentheses alpha minus gamma close parentheses end exponent over denominator cos squared space open parentheses alpha minus gamma close parentheses plus sin squared space open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator 2 times sin space open parentheses alpha plus gamma close parentheses times cos space open parentheses alpha minus gamma close parentheses over denominator cos squared space open parentheses alpha minus gamma close parentheses plus sin squared space open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator sin space open parentheses alpha plus gamma plus alpha minus gamma close parentheses plus sin space open parentheses alpha plus gamma minus open parentheses alpha minus gamma close parentheses close parentheses over denominator cos squared space open parentheses alpha minus gamma close parentheses plus sin squared space open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator sin space 2 alpha plus sin space 2 gamma over denominator cos squared space open parentheses alpha minus gamma close parentheses plus sin squared space open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator 2 times open parentheses sin space 2 alpha plus sin space 2 gamma close parentheses over denominator 2 times cos squared space open parentheses alpha minus gamma close parentheses plus 2 times sin squared space open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator 2 times open parentheses sin space 2 alpha plus sin space 2 gamma close parentheses over denominator 1 plus cos space 2 open parentheses alpha minus gamma close parentheses plus 1 minus cos space 2 open parentheses alpha plus gamma close parentheses end fraction end cell row blank equals cell fraction numerator 2 times open parentheses sin space 2 alpha plus sin space 2 gamma close parentheses over denominator 2 plus cos space open parentheses 2 alpha minus 2 gamma close parentheses minus cos space open parentheses 2 alpha plus 2 gamma close parentheses end fraction end cell row blank blank cell blank subscript text Oleh karena end text cos space left parenthesis 2 alpha minus 2 gamma right parenthesis minus cos space left parenthesis 2 alpha plus 2 gamma right parenthesis equals cos space 2 alpha times cos space 2 gamma plus sin space 2 alpha times sin space 2 gamma minus left parenthesis cos space 2 alpha times cos space 2 gamma minus sin space 2 alpha times sin space 2 gamma right parenthesis equals cos space 2 alpha times cos space 2 gamma plus sin space 2 alpha times sin space 2 gamma minus cos space 2 alpha times cos space 2 gamma plus sin space 2 alpha times sin space 2 gamma right parenthesis equals cos space 2 alpha times cos space 2 gamma minus cos space 2 alpha times cos space 2 gamma plus sin space 2 alpha times sin space 2 gamma plus sin space 2 alpha times sin space 2 gamma equals 2 times sin space 2 alpha times sin space 2 gamma text maka diperoleh end text end subscript end cell row blank blank blank row blank equals cell fraction numerator 2 times open parentheses sin space 2 alpha plus sin space 2 gamma close parentheses over denominator 2 plus 2 times sin space 2 alpha times sin space 2 gamma end fraction end cell row blank equals cell fraction numerator 2 times open parentheses sin space 2 alpha plus sin space 2 gamma close parentheses over denominator 2 times open parentheses 1 plus sin space 2 alpha times sin space 2 gamma close parentheses end fraction end cell row cell sin space 2 beta end cell equals cell fraction numerator sin space 2 alpha plus sin space 2 gamma over denominator 1 plus sin space 2 alpha times sin space 2 gamma end fraction end cell end table    

Dengan demikian, telah ditunjukan bahwa sin space 2 beta equals fraction numerator sin space 2 alpha plus sin space 2 gamma over denominator 1 plus sin space 2 alpha times sin space 2 gamma end fraction.space 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Ridha

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 23 September 2021

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