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Pertanyaan

Hitunglah tanpa menggunakan tabel trigonometri maupun kalkulator.

tan 20 degree space tan 40 degree space tan 80 degree 

S. Solehuzain

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Jawaban

hasil dari tan 20 degree space tan 40 degree space tan 80 degree adalah square root of 3.

Pembahasan

Ingat tan open parentheses alpha close parentheses equals fraction numerator sin open parentheses alpha close parentheses over denominator cos open parentheses alpha close parentheses end fraction

Sehingga diperoleh

 tan 20 degree space tan 40 degree space tan 80 degree equals fraction numerator sin 20 degree space sin 40 degree space sin 80 degree over denominator cos 20 degree space cos 40 degree space cos 80 degree end fraction

  • Mencari nilai sin 20 degree space sin 40 degree space sin 80 degree

Ingat rumus perkalian trigonometri 
2 space sin alpha space cos beta equals sin open parentheses alpha plus beta close parentheses plus sin open parentheses alpha minus beta close parentheses.

Sehingga diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin 20 degree space sin 40 degree space sin 80 degree end cell row blank equals cell sin 20 degree.1 half open parentheses 2 space space sin 80 degree space sin 40 degree close parentheses end cell row blank equals cell sin 20 degree.1 half open parentheses cos open parentheses 80 degree minus 40 degree close parentheses minus cos open parentheses 80 degree plus 40 degree close parentheses close parentheses end cell row blank equals cell sin 20 degree.1 half open parentheses cos open parentheses 40 degree close parentheses minus cos open parentheses 120 degree close parentheses close parentheses end cell row blank equals cell sin 20 degree.1 half open parentheses cos open parentheses 40 degree close parentheses minus open parentheses negative 1 half close parentheses close parentheses end cell row blank equals cell sin 20 degree.1 half open parentheses cos open parentheses 40 degree close parentheses plus 1 half close parentheses end cell row blank equals cell 1 half sin 20 degree cos open parentheses 40 degree close parentheses plus 1 fourth end cell end table

ingat 2 space sin alpha space cos beta equals sin open parentheses alpha plus beta close parentheses plus sin open parentheses alpha minus beta close parentheses

1 half sin 20 degree cos open parentheses 40 degree close parentheses plus 1 fourth sin 20 degree equals 1 fourth open parentheses 2 space sin 20 degree space cos 40 degree close parentheses plus 1 fourth sin 20 degree equals 1 fourth open parentheses sin open parentheses 20 degree plus 40 degree close parentheses plus sin open parentheses 20 degree minus 60 degree close parentheses close parentheses plus 1 fourth sin 20 degree equals 1 fourth open parentheses sin open parentheses 60 degree close parentheses plus sin open parentheses negative 20 degree close parentheses close parentheses plus 1 fourth sin 20 degree space text ingat end text space sin open parentheses negative a close parentheses equals negative sin open parentheses a close parentheses equals 1 fourth open parentheses 1 half square root of 3 minus sin open parentheses 20 degree close parentheses close parentheses plus 1 fourth sin 20 degree equals 1 over 8 square root of 3 minus 1 fourth sin open parentheses 20 degree close parentheses plus 1 fourth sin 20 degree equals 1 over 8 square root of 3

  • Mencari nilai cos 20 degree space cos 40 degree space cos 80 degree

Ingat rumus sudut rangkap trigonometri sin open parentheses 2 alpha close parentheses equals 2 sin open parentheses a close parentheses space cos open parentheses a close parentheses.

Sehingga diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses cos 20 degree space cos 40 degree space cos 80 degree space close parentheses cross times fraction numerator 2 space sin 20 degree space over denominator 2 space sin 20 degree space end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 space sin 20 degree space end fraction open parentheses 2 space sin 20 degree cos 20 degree close parentheses space cos 40 degree space cos 80 degree end cell row blank equals cell fraction numerator 1 over denominator 2 space sin 20 degree end fraction sin 40 degree space cos 40 degree space cos 80 degree end cell row blank equals cell fraction numerator 1 over denominator 2.2 space sin 20 degree end fraction 2. space sin 40 degree space cos 40 degree space cos 80 degree end cell row blank equals cell fraction numerator 1 over denominator 4 space sin 20 degree end fraction sin 80 degree space cos 80 degree end cell row blank equals cell fraction numerator 1 over denominator 2.4 space sin 20 degree end fraction 2. space sin 80 degree space cos 80 degree end cell row blank equals cell fraction numerator 1 over denominator 8 space sin 20 degree end fraction sin 160 degree end cell row blank equals cell fraction numerator 1 over denominator 8 space sin 20 degree end fraction sin open parentheses 180 degree minus 20 degree close parentheses end cell row blank equals cell fraction numerator 1 over denominator 8 up diagonal strike space sin 20 end strike degree end fraction up diagonal strike sin 20 degree end strike end cell row blank equals cell 1 over 8 end cell end table

Dari 2 perhitungan di atas kita dapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell tan 20 degree space tan 40 degree space tan 80 degree end cell equals cell fraction numerator sin 20 degree space sin 40 degree space sin 80 degree over denominator cos 20 degree space cos 40 degree space cos 80 degree end fraction end cell row blank equals cell fraction numerator up diagonal strike 1 over 8 end strike square root of 3 over denominator up diagonal strike begin display style 1 over 8 end style end strike end fraction end cell row blank equals cell square root of 3 end cell end table 

Dengan demikian, hasil dari tan 20 degree space tan 40 degree space tan 80 degree adalah square root of 3.

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