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18. Jika sin α+sin β=p dan cos α+cos β=q, buktikan: c. p2+q2​==​2{1+cos (α−β)}4 cos2 21​(α−β)​

Pertanyaan

18. Jika sin space alpha plus sin space beta equals p dan cos space alpha plus cos space beta equals q, buktikan:

c. table attributes columnalign right center left columnspacing 0px end attributes row cell p squared plus q squared end cell equals cell 2 open curly brackets 1 plus cos space open parentheses alpha minus beta close parentheses close curly brackets end cell row blank equals cell 4 space cos squared space 1 half open parentheses alpha minus beta close parentheses end cell end table 

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Pembahasan

Terlebih dahulu kita substitusi nilai p dan q pada soal tersebut, diperoleh:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell p squared plus q squared end cell equals cell open parentheses sin space alpha plus sin space beta close parentheses squared plus open parentheses cos space alpha plus cos space beta close parentheses squared end cell row blank equals cell sin squared alpha plus 2 space sin space alpha space sin space beta plus sin squared beta plus cos squared alpha plus 2 space cos space alpha space cos space beta plus cos squared beta end cell row blank equals cell open parentheses sin squared alpha plus cos squared alpha close parentheses plus open parentheses sin squared beta plus cos squared beta close parentheses plus 2 open parentheses cos space alpha space cos space beta plus sin space alpha space sin space beta close parentheses end cell end table end style 

Ingat rumus jumlah selisih sudut pada cosinus dan identitas fungsi trigonometri berikut yang akan diaplikasikan pada soal tersebut, yaitu:

  • sin squared alpha plus cos squared alpha equals 1 
  • cos open parentheses alpha minus beta close parentheses equals cos space alpha space cos space beta plus sin space alpha space sin space beta 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell p squared plus q squared end cell equals cell open parentheses sin squared alpha plus cos squared alpha close parentheses plus open parentheses sin squared beta plus cos squared beta close parentheses plus 2 open parentheses cos space alpha space cos space beta plus sin space alpha space sin space beta close parentheses end cell row blank equals cell 1 plus 1 plus 2 open parentheses cos space open parentheses alpha minus beta close parentheses close parentheses end cell row blank equals cell 2 plus 2 space cos space open parentheses alpha minus beta close parentheses end cell row blank equals cell 2 open parentheses 1 plus cos space open parentheses alpha minus beta close parentheses close parentheses end cell end table end style 

Ingat kembali identitas sudut ganda pada trigonometri yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 A end cell equals cell 2 space cos squared A space minus 1 end cell row cell cos space A end cell equals cell 2 space cos squared space 1 half A minus 1 end cell end table 

Misalkan A = open parentheses alpha minus beta close parentheses, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell p squared plus q squared end cell equals cell 2 open parentheses 1 plus cos open parentheses alpha minus beta close parentheses close parentheses end cell row blank equals cell 2 open parentheses 1 plus 2 space cos squared space 1 half open parentheses alpha minus beta close parentheses minus 1 close parentheses end cell row blank equals cell 2 open parentheses 2 space cos squared space 1 half open parentheses alpha minus beta close parentheses close parentheses end cell row blank equals cell 4 space cos squared space 1 half open parentheses alpha minus beta close parentheses end cell end table 

Dengan demikian,

terbukti bahwa Error converting from MathML to accessible text..

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