Pertanyaan

tan 20 ⋅ tan 40 ⋅ tan 80 = ....

$tan 20 \cdot tan 40 \cdot tan 80 = ....$

$- 23$
$- 3$
$1$
$3$
$23$

L. Nikmah

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Ingat identitas trigonometri tan x = cos x sin x dan rumus perkalian trigonometri berikut ini: –2 sin A sin B = cos ( A + B ) − cos ( A – B ) 2 cos A cos B = cos ( A + B ) + cos ( A – B ) 2 cos A sin B = sin ( A + B ) – sin ( A – B ) Dengan menggunakan konsep tersebut, diperoleh hasil: Jadi, tan 20 ⋅ tan 40 ⋅ tan 80 = 3 . Oleh karena itu, jawaban yang benar adalah D.

Ingat identitas trigonometri $tan x = \frac{sin x}{cos x}$ dan rumus perkalian trigonometri berikut ini:

$-2 sin A sin B = cos (A + B) - cos (A - B)$

$2 cos A cos B = cos (A + B) + cos (A - B)$

$2 cos A sin B = sin (A + B) - sin (A - B)$

Dengan menggunakan konsep tersebut, diperoleh hasil:

$tan space 20 degree space tan space 40 degree space tan space 80 degree equals fraction numerator sin space 20 degree over denominator cos space 20 degree end fraction times fraction numerator sin space 40 degree over denominator cos space 40 degree end fraction times fraction numerator sin space 80 degree over denominator cos space 80 degree end fraction equals fraction numerator sin space 20 degree blank open parentheses sin space 80 degree blank sin space 40 degree close parentheses over denominator cos space 20 degree open parentheses cos space 80 degree cos space 40 degree close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half open parentheses negative cos space open parentheses 80 degree plus 40 degree close parentheses plus cos space open parentheses 80 degree minus 40 degree close parentheses close parentheses close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses cos space open parentheses 80 degree plus 40 degree close parentheses plus cos space open parentheses 80 degree minus 40 degree close parentheses close parentheses close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half open parentheses negative cos blank 120 degree plus cos space 40 degree close parentheses close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses cos space 120 degree plus cos space 40 degree close parentheses close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 half open parentheses 1 half plus cos space 40 degree close parentheses close parentheses over denominator cos space 20 degree open parentheses 1 half open parentheses negative 1 half plus cos space 40 degree close parentheses close parentheses end fraction equals fraction numerator sin space 20 degree open parentheses 1 fourth plus 1 half blank cos space 40 degree close parentheses over denominator cos space 20 degree open parentheses negative 1 fourth plus 1 half blank cos space 40 degree close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank cos space 40 degree space sin space 20 degree over denominator negative 1 fourth cos space 20 degree plus 1 half blank cos space 40 degree cos space 20 degree end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank open parentheses 1 half open parentheses sin space open parentheses 40 degree plus 20 degree close parentheses minus sin space open parentheses 40 degree minus 20 degree close parentheses close parentheses close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 half blank open parentheses 1 half open parentheses cos space open parentheses 40 degree plus 20 degree close parentheses plus cos space open parentheses 40 degree minus 20 degree close parentheses close parentheses close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 half blank open parentheses 1 half open parentheses sin space 60 degree minus sin space 20 degree close parentheses close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 half blank open parentheses 1 half open parentheses cos space 60 degree minus cos space 20 degree close parentheses close parentheses end fraction equals fraction numerator 1 fourth sin space 20 degree plus 1 fourth blank open parentheses sin space 60 degree minus sin space 20 degree close parentheses over denominator negative 1 fourth cos space 20 degree plus 1 fourth blank open parentheses cos space 60 degree plus cos space 20 degree close parentheses end fraction equals fraction numerator sin space 20 degree plus open parentheses sin space 60 degree minus sin space 20 degree close parentheses over denominator negative cos space 20 degree plus open parentheses cos space 60 degree plus cos space 20 degree close parentheses end fraction equals fraction numerator sin space 60 degree over denominator cos space 60 degree end fraction equals tan space 60 degree equals square root of 3$

Jadi, $tan 20 \cdot tan 40 \cdot tan 80 = 3$ .

Oleh karena itu, jawaban yang benar adalah D.

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