Tanpa menggunakan tabel matematika matematika maupun kalkulator, tentukan nilai dari: b. tan 82,5∘

Pertanyaan

Tanpa menggunakan tabel matematika matematika maupun kalkulator, tentukan nilai dari:

b. $\tan\;82,5^\circ$

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

nilai dari $\tan\;82,5^\circ$ adalah $\begin{array}{rcl}&&2+\sqrt3+2\sqrt{2+\sqrt3}\end{array}$ .

Pembahasan

Ingat kembali:

-Rumus sudut paruh trigonometri:

$tan space straight A over 2 equals fraction numerator 1 minus cos space straight A over denominator sin space straight A end fraction$

$cos table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight A over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of fraction numerator 1 plus cos space straight A over denominator 2 end fraction end root end cell end table$

-ingat juga:

$\cos\;330^\circ=\cos\;30^\circ$

Pertama kita hitung nilai dari $\sin\;165^\circ$ dan $\cos\;165^\circ$ dengan menggunakan sudut paruh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A over 2 end cell equals cell plus-or-minus square root of fraction numerator 1 minus cos space straight A over denominator 2 end fraction end root end cell row cell sin space fraction numerator 330 degree over denominator 2 end fraction end cell equals cell square root of fraction numerator 1 minus cos space 330 degree over denominator 2 end fraction end root open parentheses positif space karena space kuadran space II close parentheses end cell row blank equals cell square root of fraction numerator 1 minus cos space 30 degree over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator 1 minus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator begin display style fraction numerator 2 minus square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator 2 minus square root of 3 over denominator 4 end fraction end root end cell row cell sin space 165 degree end cell equals cell 1 half square root of 2 minus square root of 3 end root end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight A over 2 end cell equals cell plus-or-minus square root of fraction numerator 1 plus cos space straight A over denominator 2 end fraction end root end cell row cell cos space fraction numerator 330 degree over denominator 2 end fraction end cell equals cell square root of fraction numerator 1 plus cos space 330 over denominator 2 end fraction end root space open parentheses negatif space karena space kuadran space II close parentheses end cell row blank equals cell square root of fraction numerator 1 plus cos space 30 over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator 1 plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator begin display style fraction numerator 2 plus square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell square root of fraction numerator 2 plus square root of 3 over denominator 4 end fraction end root end cell row cell cos space 165 end cell equals cell 1 half square root of 2 plus square root of 3 end root end cell end table$

Sehingga diperoleh perhitungan:

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space straight A over 2 end cell equals cell fraction numerator 1 minus cos space straight A over denominator sin space straight A end fraction end cell row cell tan space fraction numerator 165 degree over denominator 2 end fraction end cell equals cell fraction numerator 1 minus cos space 165 degree over denominator sin space 165 degree end fraction end cell row cell tan space 82 comma 5 degree end cell equals cell fraction numerator 1 minus open parentheses negative begin display style 1 half end style square root of 2 plus square root of 3 end root close parentheses over denominator begin display style 1 half square root of 2 minus square root of 3 end root end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 plus square root of 2 plus square root of 3 end root over denominator up diagonal strike 2 end fraction end style over denominator begin display style fraction numerator square root of 2 minus square root of 3 end root over denominator up diagonal strike 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 2 plus square root of 2 plus square root of 3 end root over denominator square root of 2 minus square root of 3 end root end fraction cross times fraction numerator square root of 2 minus square root of 3 end root over denominator square root of 2 minus square root of 3 end root end fraction end cell row blank equals cell fraction numerator 2 square root of 2 minus square root of 3 end root plus square root of open parentheses 2 plus square root of 3 close parentheses open parentheses 2 minus square root of 3 close parentheses end root over denominator 2 minus square root of 3 end fraction end cell row blank equals cell fraction numerator 2 square root of 2 minus square root of 3 end root plus square root of 1 over denominator 2 minus square root of 3 end fraction end cell row blank equals cell fraction numerator 2 square root of 2 minus square root of 3 end root plus 1 over denominator 2 minus square root of 3 end fraction cross times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction end cell row blank equals cell fraction numerator open parentheses 2 square root of 2 minus square root of 3 end root open parentheses 2 plus square root of 3 close parentheses close parentheses plus 2 plus square root of 3 over denominator 4 minus 3 end fraction end cell row blank equals cell 2 plus square root of 3 plus open parentheses 2 square root of 2 minus square root of 3 end root open parentheses 2 plus square root of 3 close parentheses close parentheses times fraction numerator square root of 2 plus square root of 3 end root over denominator square root of 2 plus square root of 3 end root end fraction end cell row blank equals cell 2 plus square root of 3 plus 2 square root of 2 plus square root of 3 end root times square root of open parentheses 2 minus square root of 3 close parentheses times open parentheses 2 plus square root of 3 close parentheses end root end cell row blank equals cell 2 plus square root of 3 plus 2 square root of 2 plus square root of 3 end root square root of 1 end cell row blank equals cell 2 plus square root of 3 plus 2 square root of 2 plus square root of 3 end root end cell end table$

Jadi, nilai dari $\tan\;82,5^\circ$ adalah $\begin{array}{rcl}&&2+\sqrt3+2\sqrt{2+\sqrt3}\end{array}$ .

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