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Jika A , B , C adalah sudut-sudut sebuah segitiga, buktikan setiap pernyataan berikut. c. tan C = tan A ⋅ tan B − 1 tan A + tan B ​

Jika  adalah sudut-sudut sebuah segitiga, buktikan setiap pernyataan berikut.

c.  

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S. Nur

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terbukti bahwa .

terbukti bahwa tan space straight C equals fraction numerator tan space straight A plus tan space straight B over denominator tan space straight A times tan space straight B minus 1 end fraction.

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Ingat kembali: Diketahui adalah sudut-sudut sebuah segitiga. Akan dibuktikan . Maka: - Ubah bentuk ruas kiri - Ubah bentuk ruas kanan Sehingga: Jadi, terbukti bahwa .

Ingat kembali:

tan space straight A equals fraction numerator sin space straight A over denominator cos space straight A end fraction tan space open parentheses straight A plus straight B close parentheses equals fraction numerator tan space straight A plus tan space straight B over denominator 1 minus tan space straight A times tan space straight B end fraction tan space open parentheses 180 degree minus straight A close parentheses equals negative tan space straight A  

Diketahui straight A comma space straight B comma space straight C adalah sudut-sudut sebuah segitiga. Akan dibuktikan tan space straight C equals fraction numerator tan space straight A plus tan space straight B over denominator tan space straight A times tan space straight B minus 1 end fraction. Maka:

- Ubah bentuk ruas kiri

table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell tan space straight C end cell row blank equals cell tan open parentheses 180 degree minus open parentheses straight A plus straight B close parentheses close parentheses end cell row blank equals cell negative tan open parentheses straight A plus straight B close parentheses end cell row blank equals cell negative open parentheses fraction numerator sin space open parentheses straight A plus straight B close parentheses over denominator cos space open parentheses straight A plus straight B close parentheses end fraction close parentheses end cell row blank equals cell negative open parentheses fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator cos space straight A times cos space straight B minus sin space straight A times sin space straight B end fraction close parentheses end cell row blank equals cell fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator negative cos space straight A times cos space straight B plus sin space straight A times sin space straight B end fraction end cell row blank equals cell fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator sin space straight A times sin space straight B minus cos space straight A times cos space straight B end fraction end cell end table  

- Ubah bentuk ruas kanan

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kanan end cell equals cell fraction numerator tan space straight A plus tan space straight B over denominator tan space straight A times tan space straight B minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space straight A over denominator cos space straight A end fraction end style plus begin display style fraction numerator sin space straight B over denominator cos space straight B end fraction end style over denominator begin display style fraction numerator sin space straight A over denominator cos space straight A end fraction end style times begin display style fraction numerator sin space straight B over denominator cos space straight B end fraction end style minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator cos space straight A times cos space straight B end fraction end style over denominator begin display style fraction numerator sin space straight A times sin space straight B minus cos space straight A times cos space straight B over denominator cos space straight A times cos space straight B end fraction end style end fraction end cell row blank equals cell fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator up diagonal strike cos space straight A times cos space straight B end strike end fraction cross times fraction numerator up diagonal strike cos space straight A times cos space straight B end strike over denominator sin space straight A times sin space straight B minus cos space straight A times cos space straight B end fraction end cell row blank equals cell fraction numerator sin space straight A times cos space straight B plus cos space straight A times sin space straight B over denominator sin space straight A times sin space straight B minus cos space straight A times cos space straight B end fraction end cell end table end style 

Sehingga:

Ruas space kiri equals Ruas space kanan 

Jadi, terbukti bahwa tan space straight C equals fraction numerator tan space straight A plus tan space straight B over denominator tan space straight A times tan space straight B minus 1 end fraction.

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