Roboguru

Nilai dari (tan10∘+tan70∘−tan50∘) adalah ....

Pertanyaan

Nilai dari open parentheses tan space 10 degree plus tan space 70 degree minus tan space 50 degree close parentheses adalah ....

  1. 1 third

  2. fraction numerator 1 over denominator square root of 3 end fraction

  3. 1

  4. square root of 3

  5. 3

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

2 space sin space A space cos space B equals sin space left parenthesis A plus B right parenthesis plus sin space left parenthesis A minus B right parenthesis

2 space cos space A space sin space B equals sin space left parenthesis A plus B right parenthesis minus sin space left parenthesis A minus B right parenthesis

2 space cos space A space cos space B equals cos space left parenthesis A plus B right parenthesis plus cos space left parenthesis A minus B right parenthesis

sin space A minus sin space B equals space 2 space cos space 1 half open parentheses A plus B close parentheses space sin space 1 half open parentheses A minus B close parentheses

cos space A plus cos space B equals 2 space cos space 1 half open parentheses A plus B close parentheses space cos space 1 half open parentheses A minus B close parentheses

Dengan menggunakan konsep di atas, diperoleh hasil:

tan space 10 degree plus tan space 70 degree minus tan space 50 degree equals open parentheses fraction numerator sin space 10 degree over denominator cos space 10 degree end fraction plus fraction numerator sin space 70 degree over denominator cos space 70 degree end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals open parentheses fraction numerator cos space 70 degree space sin space 10 degree plus sin space 70 degree space cos space 10 degree over denominator cos space 70 degree space cos space 10 degree end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals 2 over 2 times open parentheses fraction numerator cos space 70 degree space sin space 10 degree plus sin space 70 degree space cos space 10 degree over denominator cos space 70 degree space cos space 10 degree end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals open parentheses fraction numerator 2 space cos space 70 degree space sin space 10 degree plus 2 space sin space 70 degree space cos space 10 degree over denominator 2 space cos space 70 degree space cos space 10 degree end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals open parentheses fraction numerator open parentheses sin space open parentheses 70 degree plus 10 degree close parentheses minus sin space open parentheses 70 degree minus 10 degree close parentheses close parentheses plus open parentheses sin space open parentheses 70 degree plus 10 degree close parentheses plus sin space open parentheses 70 degree minus 10 degree close parentheses close parentheses over denominator cos space open parentheses 70 degree plus 10 degree close parentheses plus cos space open parentheses 70 degree minus 10 degree close parentheses end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals open parentheses fraction numerator open parentheses sin space 80 degree minus sin space 60 degree close parentheses plus open parentheses sin space 80 degree plus sin space 60 degree close parentheses over denominator cos space 80 degree plus cos space 60 degree end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals open parentheses fraction numerator 2 space sin space 80 degree over denominator cos space 80 degree plus begin display style 1 half end style end fraction close parentheses minus fraction numerator sin space 50 degree over denominator cos space 50 degree end fraction equals fraction numerator 2 space sin space 80 degree cos space 50 degree minus sin space 50 degree space open parentheses cos space 80 degree plus 1 half close parentheses over denominator cos space 80 degree space cos space 50 degree plus begin display style 1 half cos space 50 degree end style end fraction space space space space left parenthesis samakan space penyebut right parenthesis equals fraction numerator 2 space sin space 80 degree cos space 50 degree minus cos space 80 degree space sin space 50 degree minus 1 half space sin space 50 degree over denominator cos space 80 degree space cos space 50 degree plus begin display style 1 half space cos space 50 degree end style end fraction equals fraction numerator 2 space sin space 80 degree cos space 50 degree minus cos space 80 degree space sin space 50 degree minus 1 half space sin space 50 degree over denominator cos space 80 degree space cos space 50 degree space plus begin display style 1 half space cos space 50 degree end style end fraction times 2 over 2 equals fraction numerator 2 times 2 space sin space 80 degree cos space 50 degree minus 2 space cos space 80 degree space sin space 50 degree minus sin space 50 degree over denominator 2 space cos space 80 degree space cos space 50 degree plus begin display style cos space 50 degree end style end fraction equals fraction numerator 2 open parentheses sin space open parentheses 80 degree plus 50 degree close parentheses plus sin space open parentheses 80 degree minus 50 degree close parentheses close parentheses minus open parentheses sin space open parentheses 80 degree plus 50 degree close parentheses minus sin space open parentheses 80 degree minus 50 degree close parentheses close parentheses minus sin space 50 degree over denominator cos space open parentheses 80 degree plus 50 degree close parentheses plus cos space open parentheses 80 degree minus 50 degree close parentheses plus begin display style cos space 50 degree end style end fraction equals fraction numerator 2 open parentheses sin space 130 degree plus sin space 30 degree close parentheses minus open parentheses sin space 130 degree minus sin space 30 degree close parentheses minus sin space 50 degree over denominator cos space 130 degree plus cos space 30 degree plus begin display style cos space 50 degree end style end fraction equals fraction numerator 2 open parentheses sin space 130 degree plus 1 half close parentheses minus open parentheses sin space 130 degree minus 1 half close parentheses minus sin space 50 degree over denominator cos space 130 degree plus 1 half square root of 3 plus begin display style cos space 50 degree end style end fraction equals fraction numerator 2 space sin space 130 degree plus 1 minus sin space 130 degree plus 1 half minus sin space 50 degree over denominator 1 half square root of 3 plus cos space 130 degree plus begin display style cos space 50 degree end style end fraction equals fraction numerator 3 over 2 plus sin space 130 degree minus sin space 50 degree over denominator 1 half square root of 3 plus cos space 130 degree plus begin display style cos space 50 degree end style end fraction equals fraction numerator 3 over 2 plus open parentheses 2 space cos space 1 half open parentheses 130 degree plus 50 degree close parentheses space sin space 1 half open parentheses 130 degree minus 50 degree close parentheses close parentheses over denominator 1 half square root of 3 plus open parentheses 2 space cos space 1 half open parentheses 130 degree plus 50 degree close parentheses space cos space 1 half open parentheses 130 degree minus 50 degree close parentheses close parentheses end fraction equals fraction numerator 3 over 2 plus open parentheses 2 space cos space 1 half open parentheses 180 degree close parentheses space sin space 1 half open parentheses 80 degree close parentheses close parentheses over denominator 1 half square root of 3 plus open parentheses 2 space cos space 1 half open parentheses 180 degree close parentheses space cos space 1 half open parentheses 80 degree close parentheses close parentheses end fraction equals fraction numerator 3 over 2 plus open parentheses 2 space cos space 90 degree space sin space 40 degree close parentheses over denominator 1 half square root of 3 plus open parentheses 2 space cos space 90 degree space cos space 40 degree close parentheses end fraction equals fraction numerator 3 over 2 plus open parentheses 2 times 0 times sin space 40 degree close parentheses over denominator 1 half square root of 3 plus open parentheses 2 times 0 times cos space 40 degree close parentheses end fraction equals fraction numerator 3 over 2 plus 0 over denominator 1 half square root of 3 plus 0 end fraction equals fraction numerator 3 over 2 space over denominator 1 half square root of 3 space end fraction equals 3 over 2 divided by fraction numerator square root of 3 over denominator 2 end fraction equals 3 over 2 cross times fraction numerator 2 over denominator square root of 3 end fraction equals fraction numerator 3 over denominator square root of 3 end fraction equals fraction numerator 3 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction equals fraction numerator 3 square root of 3 over denominator 3 end fraction equals square root of 3

Jadi, tan space 10 degree plus tan space 70 degree minus tan space 50 degree equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table.

Oleh karena itu, jawaban yang benar adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

L. Nikmah

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 06 September 2021

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